# Trigonometry

# Trigonometry

Historic development of trigonometry

Triangles and their properties

Right triangles and trigonometric functions

Application of the trigonometric functions

Relationships between trigonometric functions

Trigonometry is a branch of applied mathematics concerned with the relationship between angles and their sides and the calculations based on them. First developed as a branch of geometry focusing on triangles during the third century BC, trigonometry was used extensively for astronomical measurements. The major trigonometric functions, including sine, cosine, and tangent, were first defined as ratios of sides in a right triangle. Since trigonometric functions are intrinsically related, they can be used to determine the dimensions of any triangle given limited information. In the eighteenth century, the definitions of trigonometric functions were broadened by being defined as points on a unit circle. This definition for the development of graphs of functions related to the angles they represent, which were periodic. Today, using the periodic nature of trigonometric functions, mathematicians and scientists have developed mathematical models to predict many natural periodic phenomena.

## Historic development of trigonometry

The word trigonometry stems from the Greek words trigonon, which means triangle, and metrein, which means to measure. It began as a branch of geometry and was utilized extensively by early Greek mathematicians to determine unknown distances. The most notable examples are the use by Aristarchus of Samos (310–230 BC) to determine the distance to the moon and the sun, and by Eratosthenes Of Cyrene (276–194 BC) to calculate Earth’s circumference. The general principles of trigonometry were formulated by Greek astronomer, Hipparchus of Nicaea (c. 166–125 BC), who is generally credited as the founder of trigonometry. His ideas were worked out by Greek mathematician and astronomer Claudius Ptolemy of Alexandria (AD c. 90–168), who used them to develop the influential Ptolemaic theory of astronomy. Much of the information scientists know about the work of Hipparchus and Ptolemy comes from Ptolemy’s compendium, The Almagest, written around 150.

Trigonometry was initially considered a field of the science of astronomy. It was later established as a separate branch of mathematics—largely through the work of Swiss mathematicians Johann Bernoulli (1667–1748) and Leonhard Euler (1707–1783).

## Angles

central to the study of trigonometry is the concept of an angle. An angle is defined as a geometric figure created by two lines drawn from the same point, known as the vertex. The lines are called the sides of an angle and their length is one defining characteristic of an angle. Another characteristic of an angle is its measurement or magnitude, which is determined by the amount of rotation, around the vertex, required to transpose one side on top of the other. If one side is rotated completely around the point, the distance traveled is known as a revolution and the path it traces is a circle.

Angle measurements are typically given in units of degrees or radians. The unit of degrees, invented by the ancient Babylonians, divides one revolution into 360° (degrees). Angles that are greater than 360° represent a magnitude greater than one revolution. Radian units, which relate angle size to the radius of the circle formed by one revolution, divide a revolution into 2π units. For most theoretical trigonometric work, the radian is the primary unit of angle measurement.

## Triangles and their properties

The principles of trigonometry were originally developed around the relationship between the sides of a triangle and its angles. The idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, whose sum is equal to 180°. The three points of a triangle, known as its vertices, are usually denoted by capital letters.

Triangles can be classified by the lengths of their sides or magnitude of their angles. Isosceles triangles have two equal sides and two congruent (equal) angles. Equilateral, or equiangular, triangles have three equal sides and angles. If no sides are equal, the triangle is a scalene triangle. All of the angles in an acute triangle are less than 90° and at least one of the angles in an obtuse triangle is greater than 90°.

Triangles, such as these, which do not contain a 90° angle, are generally known as oblique triangles. Right triangles, the most important ones to trigonometry, are those which contain one 90° angle.

Triangles, which have proportional sides and congruent angles, are called similar triangles. The concept of similar triangles, one of the basic insights in trigonometry, allows mathematicians to determine the length of a side of one triangle if they know the length of certain sides of the other triangle. For example, if one wanted to know the height of a tree, use the idea of similar triangles to find it without actually having to measure it. For example, suppose a person is 6 ft (183 cm) tall and casts an 8 ft (2.44 m) long shadow. The tree, whose height is unknown, casts a shadow that is 20 ft (6.1 m) long. The triangles that could be drawn using the shadows and objects as sides are similar. Since the sides of similar triangles are proportional, the height of the tree is determined by setting up the mathematical equality

By solving this equation, the height of the tree is found to be 15 ft (4.57 m).

## Right triangles and trigonometric functions

The triangles used in the previous example were right triangles. During the development of trigonometry, the parts of a right triangle were given certain names. The longest side of the triangle, which is directly across from the right angle, is known as the hypotenuse. The sides that form the right angle, denoted by a box in the diagram, are the legs of the triangle. For either acute angle in the triangle, the leg that forms the angle with the hypotenuse is known as the adjacent side. The side across from this angle is known as the opposite side. Typically, the length of each side is denoted by a lower case letter. In the diagram of triangle ABC, the length of the hypotenuse is indicated by c, the adjacent side is represented by b, and the opposite side by a. The angle of interest is usually represented by θ.

The ratios of the sides of a right triangle to each other are dependent on the magnitude of its acute angles. In mathematics, whenever one value depends on some other value, the relationship is known as a function. Therefore, the ratios in a right triangle are trigonometric functions of its acute angles. Since these relationships are of most importance in trigonometry, they are given special names. The ratio or number obtained by dividing the length of the opposite side by the hypotenuse is known as the sine of the angle θ (abbreviated sin θ). The ratio of the adjacent side to the hypotenuse is called the cosine of the angle θ (abbreviated cos θ). Finally, the ratio of the opposite side to the adjacent side is called the tangent of θ or tan θ. In the triangle ABC, the trigonometric functions are represented by the following equations.

These ratios represent the fundamental functions of trigonometry. Many mnemonic devices have been developed to help people remember the names of the functions and the ratios they represent. One of the easiest is the phrase SOH-CAH-TOA. This means: sine is the opposite over the hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

In addition to the three fundamental functions, three reciprocal functions are also defined. The inverse of sin θ, or 1/sin θ, is known as the secant of the angle or sec θ. The inverse of the cos θ is the cosecant or csc θ. Finally, the inverse of the tangent is called the cotangent of cot θ. These functions are typically used in special instances.

The values of the trigonometric functions can be found in various ways. They can often be looked up in trigonometric tables, which have been compiled over the years. They can also be determined by using infinite series formulas. Conveniently, most calculators and computers have the values of trigonometric functions preprogrammed inside their memory banks.

## Application of the trigonometric functions

One immediate application for trigonometric functions is the simple determination of the dimensions of a right triangle, also known as the solution of a triangle, when only a few are known. For example, if the sides of a right triangle are known, then the magnitude of both acute angles can be found. Suppose mathematicians have a right triangle whose sides are 2 in (5 cm) and 4.7 in (12 cm), and whose hypotenuse is 5.1 in (13 cm). The unknown angles could be found by using any trigonometric function. Since the sine of one of the angles is equal to the length of the opposite side divided by the hypotenuse, this angle can be determined. The sine of one angle is 5/13, or 0.385. With the help of a trigonometric function table or calculator, it will be found that the angle which has a sine of 0.385 is 22.6°. Using the fact that the sum of the angles in a triangle is 180°, one can establish that the other angle is 180° - 90° 22.6° =67.4°.

In addition to solving a right triangle, trigonometric functions can also be used in the determination of the area when given only limited information. The standard method of finding the area of a triangle is by using the formula, area = 1/2b (base) × h (altitude). Often, the altitude of a triangle is not known, but the sides and an angle are known. Using the side-angle-side (SAS) theorem, the formula for the area of a triangle then becomes, area = 1/2 (one side) × (another side) × (sine of the included angle). For a triangle with sides of 5 cm and 3 cm, respectively, and an included angle of 60°, the area of the triangle would be equal to 1/2 × 5 × 3 × sin 60° =13cm^{2}.

The formula for the area of a triangle leads to an important concept in trigonometry known as the Law of Sines, which says that for any triangle, the sine of each angle is proportional to its opposite side, symbolically written in triangle ABC as,

Using the Law of Sines, one can solve any triangle if the length of one side and magnitude of two angles, or two sides and one angle, is known. Suppose one has a triangle with angles of 45° and 70°, and an included side of 15.7 in (40 cm). The third angle is found to be 180° -45° -70° =65°. The unknown sides, x and y, are found with the Law of Sines because

The lengths of the unknown sides are then x = 12.29 in (31.2 cm) and y = 16.35 in (41.5 cm).

The Law of Sines cannot be used to solve a triangle unless at least one angle is known. However, a triangle can be solved if only the sides are known by using the Law of Cosines, which is stated in triangle ABC, c^{2} =a^{2} +b^{2} - 2ab cos C, or can be written

which is more convenient when using only the sides to solve a triangle. As an example, consider a triangle with sides equal to 2 in, 3.5 in, and 3.9 in (5 cm, 9 cm, and 10 cm). The cosine of one angle would be equal to (5^{2} + 9^{2} - 10^{2})/(2 × 59) = 0.067, which corresponds to the angle 86.2°. Similarly, the other two angles are found to be 29.9° and 63.9°.

## Relationships between trigonometric functions

In addition to the reciprocal relationships of certain trigonometric functions, two other types of relationships exist. These relationships, known as trigonometric identities, include co-functional relationships and Pythagorean relationships. Co-functional relationships relate functions by their complementary angles. Pythagorean relationships relate functions by application of the Pythagorean theorem.

The sine and cosine of an angle are considered co-functions, as are the secant and cosecant, and the tangent and cotangent.

The Pythagorean theorem states that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. For a triangle with sides of x and y and a hypotenuse of z, the equation for the Pythagorean Theorem is x^{2}+ y^{2} = z^{2}. Applying this theorem to the trigonometric functions of an angle, one finds that sin^{2} θ +cos^{2} θ =1. Similarly, 1+tan^{2} θ = sec^{2} θ and 1 + cot^{2} θ =csc^{2} θ. The terms such as sin^{2} θ or tan^{2} θ traditionally have meant (sin θ) × (sin θ) or (tan θ) × (tan θ).

In some instances, it is desirable to know the trigonometric function of the sum or difference of two angles. If one has two unknown angles, θ and θ, then sin (θ + θ) is equal to sin θ cos θ +cos θ sin θ.In a similar manner, their difference, sin(θ - θ) is sin θ cos θ - cos θ sin θ. Equations for determining the sum or differences of the cosine and tangent also exist and can be stated as follows:

cos(θ ± Ø)=cos θ cos θ ± sin θ sin θ tan (θ ± θ)= (tan θ ± tan θ)/(1 ± tan θ tan Ø).

These relationships can be used to develop formulas for double angles and half angles. Therefore, the sin 2θ = 2sin θ cos θ and cos 2θ = 2cos^{2} θ - 1, which could also be written cos θ2 = 1 - 2sin^{2} θ.

## Trigonometry using circles

For hundreds of years, trigonometry was only considered useful for determining sides and angles of a triangle. However, when mathematicians developed more general definitions for sine, cosine, and tangent, trigonometry became much more important in mathematics and science alike. The general definitions for the trigonometric functions were developed by considering these values as points on a unit circle.

A unit circle is one that has a radius of one unit, which means x^{2} +y^{2} = 1. If one considers the circle to represent the rotation of a side of an angle, then the trigonometric functions can be defined by the x and y coordinates of the point of rotation. For example, coordinates of point P(x,y) can be used to define a right triangle with a hypotenuse of length r. The trigonometric functions could then be represented by the following equations.

With the trigonometric functions defined as such, a graph of each can be developed by plotting its value versus the magnitude of the angle it represents.

Since the value for x and y can never be greater than one on a unit circle, the range for the sine and cosine graphs is between 1 and -1. The magnitude of an

### KEY TERMS

**Adjacent side** —The side of a right triangle that forms one side of the angle in question.

**Amplitude** —A characteristic of a periodic graph represented by half the distance between its maximum and minimum.

**Angle** —A geometric figure created by two lines drawn from the same point.

**Cosine** —A trigonometric function that relates the ratio of the adjacent side of a right triangle to its hypotenuse, or the x coordinate of a point on a unit circle.

**Degree** —A unit of measurement used to describe the amount of revolution of an angle denoted by the symbol^{°}. There are 360° in a complete revolution.

**Hypotenuse** —The longest side of a right triangle that is opposite the right angle.

**Law of cosines** —A relationship between the cosine of an angle of a triangle and its sides that can be used to determine the dimensions of a triangle.

**Law of sines** —A relationship between the sine of an angle of a triangle and its side that can be used to determine the dimensions of a triangle.

**Opposite side** —The side of a right triangle which is opposite the angle in question.

**Period** —A value at which a periodic function begins to repeat.

**Pythagorean theorem** —An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is used to find the distance between two points.

**Radian** —A unit of angular measurement that relates the radius of a circle to the amount of rotation of an angle. One complete revolution is equal to 2π radians.

**Right triangle** —A triangle that contains a 90° or right angle

**Similar triangles** —Triangles that have congruent angles and proportional sides.

**Sine** —A trigonometric function that represents the ratio of the opposite side of a right triangle to its hypotenuse, or the y coordinate of a point on a unit circle.

**Tangent** —A trigonometric function that represents the ratio of the opposite side of right triangle to its adjacent side.

**Trigonometric functions** —Angular functions that can be described as ratios of the sides of a right triangle to each other.

angle can be any real number, so the domain of the graphs is all real numbers. (Angles that are greater than 360° or 2π radians represent an angle with more than one revolution of rotation). The sine and cosine graphs are periodic because they repeat their values, or have a period, every 360° or 2π radians. They also have an amplitude of one that is defined as half the difference between the maximum (1) and minimum (-1) values.

Graphs of the other trigonometric functions are possible. Of these, the most important is the graph of the tangent function. Like the sine and cosine graphs, the tangent function is periodic, but it has a period of 180° or π radians. Since the tangent is equal to y/x, its range is -∞ to ∞ and its amplitude is ∞.

The periodicity of trigonometric functions is more important to modern trigonometry than the ratios they represent. Mathematicians and scientists are now able to describe many types of natural phenomena that reoccur periodically with trigonometric functions. For example, the times of sunsets, sunrises, and comets can all be calculated thanks to trigonometric functions. Also, they can be used to describe seasonal temperature changes, the movement of waves in the ocean, and even the quality of a musical sound.

## Resources

### BOOKS

Blitzer, Robert et al. *Algebra and Trigonometry.* 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 2003.

Larson, Ron. *Calculus With Analytic Geometry.* Boston: Houghton Mifflin College, 2002.

Smith, Karl J. *Essentials of Trigonometry.* Belmont, CA: Thomson Brooks/Cole, 2006.

Stewart, James, et al. *Trigonometry* Pacific Grove, CA: Brooks/Cole, 2003.

Weisstein, Eric W. The CRC *Concise Encyclopedia of Mathematics.* New York: Boca Raton, FL: Chapman & Hall/CRC, 2003.

Perry Romanowski

# Trigonometry

# Trigonometry

Trigonometry is a branch of applied **mathematics** concerned with the relationship between angles and their sides and the calculations based on them. First developed as a branch of **geometry** focusing on triangles during the third century b.c., trigonometry was used extensively for astronomical measurements. The major trigonometric functions, including sine, cosine, and tangent, were first defined as ratios of sides in a right triangle. Since trigonometric functions are intrinsically related, they can be used to determine the dimensions of any triangle given limited information. In the eighteenth century, the definitions of trigonometric functions were broadened by being defined as points on a unit **circle** . This allowed the development of graphs of functions related to the angles they represent which were periodic. Today, using the periodic nature of trigonometric functions, mathematicians and scientists have developed mathematical models to predict many natural periodic phenomena.

## Historic development of trigonometry

The word trigonometry stems from the Greek words *trigonon*, which means triangle, and *metrein*, which means to measure. It began as a branch of geometry and was utilized extensively by early Greek mathematicians to determine unknown distances. The most notable examples are the use by Aristarchus (310-250 b.c.) to determine the **distance** to the **Moon** and **Sun** , and by Eratosthenes (c. 276-195 b.c.) to calculate the Earth's circumference. The general principles of trigonometry were formulated by the Greek astronomer, Hipparchus of Nicaea (162-127 b.c.), who is generally credited as the founder of trigonometry. His ideas were worked out by Ptolemy of Alexandria (a.d. c. 90-168), who used them to develop the influential Ptolemaic theory of **astronomy** . Much of the information we know about the work of Hipparchus and Ptolemy comes from Ptolemy's compendium, *The Almagest*, written around 150.

Trigonometry was initially considered a **field** of the science of astronomy. It was later established as a separate branch of mathematics—largely through the work of the mathematicians Johann Bernoulli (1667-1748) and Leonhard Euler (1707-1783).

## Angles

Central to the study of trigonometry is the concept of an **angle** . An angle is defined as a geometric figure created by two lines drawn from the same point, known as the vertex. The lines are called the sides of an angle and their length is one defining characteristic of an angle. Another characteristic of an angle is its measurement or magnitude, which is determined by the amount of **rotation** , around the vertex, required to transpose one side on top of the other. If one side is rotated completely around the point, the distance travelled is known as a revolution and the path it traces is a circle.

Angle measurements are typically given in units of degrees or radians. The unit of degrees, invented by the ancient Babylonians, divides one revolution into 360° (degrees). Angles which are greater than 360° represent a magnitude greater than one revolution. Radian units, which relate angle size to the radius of the circle formed by one revolution, divide a revolution into 2π units. For most theoretical trigonometric work, the radian is the primary unit of angle measurement.

## Triangles and their properties

The principles of trigonometry were originally developed around the relationship between the sides of a triangle and its angles. The idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, whose sum is equal to 180°. The three points of a triangle, known as its vertices, are usually denoted by capital letters.

Triangles can be classified by the lengths of their sides or magnitude of their angles. Isosceles triangles have two equal sides and two congruent (equal) angles. Equilateral, or equiangular, triangles have three equal sides and angles. If no sides are equal, the triangle is a scalene triangle. All of the angles in an acute triangle are less than 90° and at least one of the angles in an obtuse triangle is greater than 90°. Triangles, such as these, which do not contain a 90° angle, are generally known as oblique triangles. Right triangles, the most important ones to trigonometry, are those which contain one 90°angle.

Triangles which have proportional sides and congruent angles are called similar triangles. The concept of similar triangles, one of the basic insights in trigonometry, allows us to determine the length of a side of one triangle if we know the length of certain sides of the other triangle. For example, if we wanted to know the height of a **tree** , we could use the idea of similar triangles to find it without actually having to measure it. Suppose a person is 6 ft (183 cm) tall and casts an 8 ft (2.44 m) long shadow. The tree, whose height is unknown, casts a shadow that is 20 ft (6.1 m) long. The triangles that could be drawn using the shadows and objects as sides are similar. Since the sides of a similar triangles are proportional, the height of the tree is determined by setting up the mathematical equality

By solving this equation, the height of the tree is found to be 15 ft (4.57 m).

## Right triangles and trigonometric functions

The triangles used in the previous example were right triangles. During the development of trigonometry, the parts of a right triangle were given certain names. The longest side of the triangle, which is directly across from the right angle, is known as the hypotenuse. The sides that form the right angle, denoted by a box in the diagram, are the legs of the triangle. For either acute angle in the triangle, the leg that forms the angle with the hypotenuse is known as the adjacent side. The side across from this angle is known as the opposite side. Typically, the length of each side is denoted by a lower case letter. In the diagram of triangle ABC, the length of the hypotenuse is indicated by c, the adjacent side is represented by b, and the opposite side by a. The angle of interest is usually represented by θ.

The ratios of the sides of a right triangle to each other are dependent on the magnitude of its acute angles. In mathematics, whenever one value depends on some other value, the relationship is known as a **function** . Therefore, the ratios in a right triangle are trigonometric functions of its acute angles. Since these relationships are of most importance in trigonometry, they are given special names. The **ratio** or number obtained by dividing the length of the opposite side by the hypotenuse is known as the sine of the angle θ (abbreviated sin θ). The ratio of the adjacent side to the hypotenuse is called the cosine of the angle θ (abbreviated cos θ). Finally, the ratio of the opposite side to the adjacent side is called the tangent of θ or tan θ. In the triangle ABC, the trigonometric functions are represented by the following equations.

These ratios represent the fundamental functions of trigonometry and should be committed to **memory** . Many mnemonic devices have been developed to help people remember the names of the functions and the ratios they represent. One of the easiest is the phrase "SOH-CAH-TOA." This means: sine is the opposite over the hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent.

In addition to the three fundamental functions, three **reciprocal** functions are also defined. The inverse of sin θ, or 1/sin θ, is known as the secant of the angle or sec θ. The inverse of the cos θ is the cosecant or csc θ. Finally, the inverse of the tangent is called the cotangent of cot θ. These functions are typically used in special instances.

The values of the trigonometric functions can be found in various ways. They can often be looked up in tables, which have been compiled over the years. They can also be determined by using infinite series formulas. Conveniently, most calculators and computers have the values of trigonometric functions preprogrammed in.

## Application of the trigonometric functions

One immediate application for trigonometric functions is the simple determination of the dimensions of a right triangle, also known as the solution of a triangle, when only a few are known. For example, if the sides of a right triangle are known, then the magnitude of both acute angles can be found. Suppose we have a right triangle whose sides are 2 in (5 cm) and 4.7 in (12 cm), and whose hypotenuse is 5.1 in (13 cm). The unknown angles could be found by using any trigonometric function. Since the sine of one of the angles is equal to the length of the opposite side divided by the hypotenuse, this angle can be determined. The sine of one angle is 5/13, or 0.385. With the help of a trigonometric function table or **calculator** , it will be found that the angle which has a sine of 0.385 is 22.6°. Using the fact that the sum of the angles in a triangle is 180°, we can establish that the other angle is 180° - 90° - 22.6° = 67.4°.

In addition to solving a right triangle, trigonometric functions can also be used in the determination of the area when given only limited information. The standard method of finding the area of a triangle is by using the formula, area = 1/2b (base) × h (altitude). Often, the altitude of a triangle is not known, but the sides and an angle are known. Using the side-angle-side (SAS) **theorem** , the formula for the area of a triangle then becomes, area = 1/2 (one side) × (another side) × (sine of the included angle). For a triangle with sides of 5 cm and 3 cm respectively and an included angle of 60°, the area of the triangle would be equal to 1/2 × 5 × 3 × sin 60° = 13 cm2.

The formula for the area of a triangle leads to an important concept in trigonometry known as the Law of Sines which says that for any triangle, the sine of each angle is proportional to the opposite its opposite side, symbolically written in triangle ABC as,

Using the Law of Sines, we can solve any triangle if we know the length of one side and magnitude of two angles, or two sides and one angle. Suppose we have a triangle with angles of 45° and 70°, and an included side of 15.7 in (40 cm). The third angle is found to be 180° 45° - 70° = 65°. The unknown sides, x and y, are found with the Law of Sines because

The lengths of the unknown sides are then x = 12.29 in (31.2 cm) and y = 16.35 in (41.5 cm).

The Law of Sines can not be used to solve a triangle unless at least one angle is known. However, a triangle can be solved if only the sides are known by using the Law of Cosines which is stated in triangle ABC, c2 = a2 + b2 - 2ab cos C, or can be written

which is more convenient when using only the sides to solve a triangle. As an example, consider a triangle with sides equal to 2 in, 3.5 in, and 3.9 in (5 cm, 9 cm, and 10 cm). The cosine of one angle would be equal to (52 + 92 - 102)/(2 × 59) = 0.067, which corresponds to the angle 86.2°. Similarly, the other two angles are found to be 29.9°and 63.9°.

## Relationships between trigonometric functions

In addition to the reciprocal relationships of certain trigonometric functions, two other types of relationships exist. These relationships, known as trigonometric identities, include cofunctional relationships and Pythagorean relationships. Cofunctional relationships relate functions by their complementary angles. Pythagorean relationships relate functions by application of the **Pythagorean theorem** .

The sine and cosine of an angle are considered cofunctions, as are the secant and cosecant, and the tangent and cotangent.

The Pythagorean theorem states that the sum of the squares of the sides of a right triangle is equal to the **square** of the hypotenuse. For a triangle with sides of x and y and a hypotenuse of z, the equation for the Pythagorean Theorem is x2+ y2 = z2. Applying this theorem to the trigonometric functions of an angle, we find that sin2 θ + cos2 θ = 1. Similarly, 1 + tan2 θ = sec2 θ and 1 + cot2 θ = csc2 θ . The terms such as sin2 θ or tan2 θ traditionally have meant (sin θ) × (sin θ) or (tan θ) × (tan θ).

In some instances, it is desirable to know the trigonometric function of the sum or difference of two angles. If we have two unknown angles, θ and φ, then sin ( θ + φ) is equal to sin θcos φ + cos θsin φ. In a similar manner, their difference, sin( θ- φ) is sin θcos φ cos θsin φ. Equations for determining the sum or differences of the cosine and tangent also exist and can be stated as follows:

cos( θ ± φ ) = cos θcos φ ± sin θsin φ tan ( θ ± φ ) = (tan θ ± tan φ)/(1 ± tan θtan φ)

These relationships can be used to develop formulas for double angles and half angles. Therefore, the sin 2 θ = 2sin θcos θ and cos 2 θ = 2cos2 θ - 1 which could also be written cos θ2 = 1 - 2sin2 θ.

## Trigonometry using circles

For hundreds of years, trigonometry was only considered useful for determining sides and angles of a triangle. However, when mathematicians developed more general definitions for sine, cosine and tangent, trigonometry became much more important in mathematics and science alike. The general definitions for the trigonometric functions were developed by considering these values as points on a unit circle.

A unit circle is one which has a radius of one unit which means x2 + y2 = 1. If we consider the circle to represent the rotation of a side of an angle, then the trigonometric functions can be defined by the x and y coordinates of the point of rotation. For example, coordinates of point P(x,y) can be used to define a right triangle with a hypotenuse of length r. The trigonometric functions could then be represented by the following equations.

With the trigonometric functions defined as such, a graph of each can be developed by plotting its value versus the magnitude of the angle it represents.

Since the value for x and y can never be greater than one on a unit circle, the range for the sine and cosine graphs is between 1 and -1. The magnitude of an angle can be any real number, so the **domain** of the graphs is all **real numbers** . (Angles which are greater than 360° or 2π radians represent an angle with more than one revolution of rotation). The sine and cosine graphs are periodic because they repeat their values, or have a period, every 360° or 2π radians. They also have an amplitude of one which is defined as half the difference between the maximum (1) and minimum (-1) values.

Graphs of the other trigonometric functions are possible. Of these, the most important is the graph of the tangent function. Like the sine and cosine graphs, the tangent function is periodic, but it has a period of 180° or π radians. Since the tangent is equal to y/x, its range is - ∞ to ∞ and its amplitude is ∞.

The periodicity of trigonometric functions is more important to modern trigonometry than the ratios they represent. Mathematicians and scientists are now able to describe many types of natural phenomena which reoccur periodically with trigonometric functions. For example, the times of sunsets, sunrises, and **comets** can all be calculated thanks to trigonometric functions. Also, they can be used to describe seasonal **temperature** changes, the movement of waves in the **ocean** , and even the quality of a musical sound.

## Resources

### books

Barnett, Raymond A., Michael Zeigler, Karl Byleen, and Steven Heath. *Analytic Trigonometry with Applications.* 7th ed. New York: John Wiley & Sons, 1998.

Blitzer, Robert et al. *Algebra and Trigonometry.* 2nd ed. Englewood Cliffs, NJ: Prentice Hall, 2003.

Larson, Ron. *Calculus With Analytic Geometry.* Boston: Houghton Mifflin College, 2002.

Stewart, James, et al. *Trigonometry* Pacific Grove, CA: Brooks/Cole, 2003.

Weisstein, Eric W. *The CRC Concise Encyclopedia of Mathematics.* New York: CRC Press, 1998.

Perry Romanowski

## KEY TERMS

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

**Adjacent side**—The side of a right triangle which forms one side of the angle in question.

**Amplitude**—A characteristic of a periodic graph represented by half the distance between its maximum and minimum.

**Angle**—A geometric figure created by two lines drawn from the same point.

**Cosine**—A trigonometric function that relates the ratio of the adjacent side of a right triangle to its hypotenuse, or the x coordinate of a point on a unit circle.

**Degree**—A unit of measurement used to describe the amount of revolution of an angle denoted by the symbol °. There are 360° in a complete revolution.

**Hypotenuse**—The longest side of a right triangle which is opposite the right angle.

**Law of cosines**—A relationship between the cosine of an angle of a triangle and its sides which can be used to determine the dimensions of a triangle.

**Law of sines**—A relationship between the sine of an angle of a triangle and its side which can be used to determine the dimensions of a triangle.

**Opposite side**—The side of a right triangle which is opposite the angle in question.

**Period**—A value at which a periodic function begins to repeat.

**Pythagorean theorem**—An idea suggesting that the sum of the squares of the sides of a right triangle is equal to the square of the hypotenuse. It is used to find the distance between two points.

**Radian**—A unit of angular measurement that relates the radius of a circle to the amount of rotation of an angle. One complete revolution is equal to 2π radians.

**Right triangle**—A triangle which contains a 90° or right angle

**Similar triangles**—Triangles which have congruent angles and proportional sides.

**Sine**—A trigonometric function which represents the ratio of the opposite side of a right triangle to its hypotenuse, or the y coordinate of a point on a unit circle.

**Tangent**—A trigonometric function which represents the ratio of the opposite side of right triangle to its adjacent side.

**Trigonometric functions**—Angular functions which can be described as ratios of the sides of a right triangle to each other.

# Trigonometry

# Trigonometry

The word "trigonometry" comes from two Greek words meaning "triangle measure." Trigonometry concerns the relationships among the sides and angles of triangles. It also concerns the properties and applications of these relationships, which extend far beyond triangles to real-world problems.

Evidence of a knowledge of elementary trigonometry dates to the ancient Egyptians and Babylonians. Led by Ptolemy, the Greeks added to this field of knowledge during the first millennium b.c.e.; simultaneously, similar work was produced in India. Around 1000 c.e., Muslim astronomers made great advances in trigonometry. Inspired by advances in astronomy, Europeans contributed to the development of this important mathematical area from the twelfth century until the days of Leonhard Euler in the eighteenth century.

## Trigonometric Ratios

To understand the six trigonometric functions, consider right triangle *ABC* with right angle *C.* Although triangles with identical angle measures may have sides of different lengths, they are similar. Thus, the ratios of the corresponding sides are equal. Because there are three sides, there are six possible ratios.

Working from angle *A,* label the sides as follows: side *c* represents the **hypotenuse** ; leg *a* represents the side opposite angle *A* ; and leg *b* is adjacent to angle *A.* The definitions of the six trigonometric functions of angle *A* are listed below.

For any angle **congruent** to angle *A,* the numerical value of any of these ratios will be equal to the value of that ratio for angle *A.* Consequently, for any given angle, these ratios have specific values that are listed in tables or can be found on calculators.

**Basic Uses of Trigonometry.** The definitions of the six functions and the Pythagorean Theorem provide a powerful means of finding unknown sides and angles. For any right triangle, if the measures of one side and either another side or angle are known, the measures of the other sides and angles can be determined.

For example, suppose the measure of angle *A* is 36° and side *c* measures 12 centimeters (and angle *C* measures 90°). To determine the measure of angle *B,* subtract 36 from 90 because the two non-right angles must sum to 90°. To determine sides *a* and *b,* solve the equations sin and cos , keeping in mind that sin 36° and cos 36° have number values. The results are *a* = 12sin 36° ≈ 7.1 cm and *b* = 12cos 36° ≈ 9.7 cm.

Two theorems that are based on right-triangle trigonometry—the Law of Sines and the Law of Cosines—allow us to solve for the unknown parts of any triangle, given sufficient information. The two laws, which can be expressed in various forms, follow.

**Law of Cosines:** *a ^{2}* =

*b*+

^{2}*c*−

^{2}*2bc*cos

*A*

## Expanded Uses of Trigonometry

The study of trigonometry goes far beyond just the study of triangles. First, the definitions of the six trigonometric functions must be expanded. To accomplish this, establish a rectangular coordinate system with *P* at the origin. Construct a circle of any radius, using point *P* as the center. The positive horizontal axis represents 0°. As one moves counter-clockwise along the circle, a positive angle *A* is generated.

Consider a point on the circle with coordinates (*u, v* ). (The reason for using the coordinates (*u, v* ) instead of (*x, y* ) is to avoid confusion later on when constructing graphs such as *y* = sin *x.* ) By projecting this point onto the horizontal axis as shown below, a direct analogy to the original trigonometric functions can be made. The length of the adjacent side equals the *u* -value, the length of the opposite side equals the *v* -value, and the length of the hypotenuse equals the radius of the circle. Thus the six trigonometric functions are expanded because they are no longer restricted to **acute** angles.

For any circle, similar triangles are created for equal central angles. Consequently, one can choose whatever radius is most convenient. To simplify calculations, a circle of radius 1 is often chosen. Notice how four of the functions, especially the sine and cosine functions, become much simpler if the radius is 1.

These expanded definitions, which relate an angle to points on a circle, allow for the use of trigonometric functions for any angle, regardless of size. So far the angles discussed have been measured in degrees. This, however, limits the applicability of trigonometry. Trigonometry is far less restricted if angles are measured in units called **radians** .

**Using Radian Measure.** Because all circles are similar, for a given central angle in any circle, the ratio of an intercepted arc to the radius is constant. Consequently, this ratio can be used instead of the degree measure to indicate the size of an angle.

Consider for example a semicircle with radius 4 centimeters. The arc length, which is half of the circumference, is exactly 4π centimeters. In radians, therefore, the angle is the ratio 4π centimeters to 4 centimeters, or simply π. (There are no units when radian measure is used.) This central angle also measures 180°. Recognizing that 180° is equivalent to π (when measured in radians), there is now an easy way of converting to and from degrees and radians. This can also be used to determine that an angle of 1 radian, an angle which intercepts an arc that is precisely equal to the radius of the circle, is approximately 57.3°.

Now the domain for the six trigonometric functions may be expanded beyond angles to the entire set of real numbers. To do this, define the trigonometric function of a number to be equivalent to the same function of an angle measuring that number of radians. For example, an expression such as sin 2 is equivalent to taking the sine of an angle measuring 2 radians. With this freedom, the trigonometric functions provide an excellent tool for studying many real-world phenomena that are periodic in nature.

The figure below shows the graphs of the sine, cosine, and tangent functions, respectively. Except for values for which the tangent is undefined, the domain for these functions is the set of real numbers. The domain for the parts of the graphs that are shown is −2π ≤ *x* ≤ 2π. Each tick mark on the *x* -axis represents units, and each tick mark on the *y* -axis represents one unit.

To understand the graphs, think back to a circle with radius 1. Because the radius is 1, the sine function, which is defined as , simply traces the vertical value of a point as it moves along the circumference of the circle. It starts at 0, moves up as high as 1 when the angle is (90°), retreats to 0, goes down to −1, returns to 0, and begins over again. The graph of the cosine function is identical except for being (90°) out of phase. It records the horizontal value of a point as it moves along the unit circle.

The tangent is trickier because it concerns the ratio of the vertical value to the horizontal value. Whenever the vertical component is 0, which happens at points along the horizontal axis, the tangent is 0. Whenever the horizontal component is 0, which happens at points on the vertical axis, the tangent is not defined—or infinite. Thus, the tangent has a vertical **asymptote** every π units.

**A Practical Example.** By moving the sine, cosine, and tangent graphs left or right and up or down and by stretching them horizontally and vertically, these trigonometric functions serve as excellent models for many things. For example, consider the function, in which *x* represents the month of the year and , in which *x* represents the average monthly temperature measured in Fahrenheit.

The "parent" function is the cosine, which intercepts the vertical axis at its maximum value. In our model, we find the maximum value shifted 7 units to the right, indicating that the maximum temperature occurs in the seventh month, July.

The normal period of the cosine function is 2π units, but our transformed function is only going as fast, telling us that it takes 12 units, in this case months, to complete a cycle.

The amplitude of the parent graph is 1; this means that its highest and lowest points are both 1 unit away from its horizontal axis, which is the mean functional (vertical) value. In our example, the amplitude is 22, indicating that its highest point is 22 units (degrees) above its average and its lowest is 22 degrees below its average. Thus, there is a 44-degree difference between the average temperature in July and the average temperature in January, which is half a cycle away from July.

Finally, the horizontal axis of the parent function is the *x* -axis; in other words, the average height is 0. In this example, the horizontal average has been shifted up 54 units. This indicates that the average spring temperature—in April to be specific—is 54 degrees. So too is the average temperature in October. Combining this with the amplitude, it is found that the average July temperature is 76 degrees, and the average January temperature is 32 degrees.

Trigonometric equations often arise from these mathematical models. If, in the previous example, one wants to know when the average temperature is 65 degrees, 65 is substituted for *y,* and the equation is solved for *x.* Any of several techniques, including the use of a graph, can work. Similarly, if one wishes to know the average temperature in June, 6 is substituted for *x,* and the equation is solved for *y.*

see also Angles, Measurement of.

*Bob Horton*

## Bibliography

Boyes, G. R. "Trigonometry for Non-Trigonometry Students." *Mathematics Teacher* 87, no. 5 (1994): 372–375.

Klein, Raymond J., and Ilene Hamilton. "Using Technology to Introduce Radian Measure." *Mathematics Teacher* 90, no. 2 (1997): 168–172.

Peterson, Blake E., Patrick Averbeck, and Lynanna Baker. "Sine Curves and Spaghetti." *Mathematics Teacher* 91, no. 7 (1998): 564–567.

Swetz, Frank J., ed. *From Five Fingers to Infinity: A Journey through the History of Mathematics.* Chicago: Open Court Publishing Company, 1994.

# Trigonometry

# Trigonometry

Trigonometry is a branch of mathematics concerned with the relationship between angles and their sides and the calculations based on them. First developed during the third century b.c. as a branch of geometry focusing on triangles, trigonometry was used extensively for astronomical measurements. The major trigonometric functions—including sine, cosine, and tangent—were first defined as ratios of sides in a right triangle. Since trigonometric functions are a natural part of any triangle, they can be used to determine the dimensions of any triangle given limited information.

In the eighteenth century, the definitions of trigonometric functions were broadened by being defined as points on a unit circle. This development allowed the construction of graphs of functions related to the angles they represent, which were periodic. Today, using the periodic (regularly repeating) nature of trigonometric functions, mathematicians and scientists have developed mathematical models to predict many natural periodic phenomena.

## Trigonometric functions

The principles of trigonometry were originally developed around the relationship among the sides of a right triangle and its angles. The basic idea was that the unknown length of a side or size of an angle could be determined if the length or magnitude of some of the other sides or angles were known. Recall that a triangle is a geometric figure made up of three sides and three angles, the sum of the angles equaling 180 degrees. The three points of a triangle, known as its vertices, are usually denoted by capital letters.

## Words to Know

**Adjacent side:** The side of a right triangle that forms one side of the angle in question.

**Angle:** A geometric figure created by two lines drawn from the same point.

**Cosine:** A trigonometric function that relates the ratio of the adjacent side of a right triangle to its hypotenuse.

**Geometry:** A branch of mathematics originally developed and used to measure common features on Earth, such as lines, circles, angles, triangles, squares, trapezoids, spheres, cones, and cylinders.

**Hypotenuse:** The longest side of a right triangle that is opposite the right angle.

**Opposite side:** The side of a right triangle that is opposite the angle in question.

**Periodic function:** A function that changes regularly over time.

**Radian:** A unit of angular measurement that relates the radius of a circle to the amount of rotation of the angle. One complete revolution is equal to 2*π* radians.

**Right triangle:** A triangle that contains a 90-degree or right angle.

**Sine:** A trigonometric function that represents the ratio of the opposite side of a right triangle to its hypotenuse.

**Tangent:** A trigonometric function that represents the ratio of the opposite side of right triangle to its adjacent side.

**Trigonometric function:** An angular function that can be described as the ratio of the sides of a right triangle to each other.

**Vertices:** The point where two lines come together, such as the corners of a triangle.

The longest side of a right triangle, which is directly across the right angle, is known as the hypotenuse. The sides that form the right angle are the legs of the triangle. For either acute angle (less than 90 degrees) in the triangle, the leg that forms the angle with the hypotenuse is known as the adjacent side. The side across from this angle is known as the opposite side. Typically, the length of each side of the right triangle is denoted by a lowercase letter.

Three basic functions—the sine (sin), cosine (cos), and tangent (tan)—can be defined for any right triangle. Those functions are defined as follows:

sin *θ* = length of opposite side *÷* length of hypotenuse, or ^{a}/_{c}

cos *θ* = length of adjacent side *÷* length of hypotenuse, or ^{b}/_{c}

tan *θ* = length of opposite side *÷* length of adjacent side, or ^{a}/_{b}

Three other functions—the secant (sec), cosecant (csc), and cotangent (cot)—can be derived from these three basic functions. Each is the inverse of the basic function. Those inverse functions are as follows:

sec *θ* = 1/sin *θ* = c/a

csc *θ* = 1/cos *θ* = c/b

cot *θ* = 1/tan *θ* = b/a

## Periodicity of trigonometric functions

One of the most useful characteristics of trigonometric functions is their periodicity. The term periodicity means that the function repeats itself over and over again in a very regular fashion. For example, suppose that you graph the function y = sin *θ*. In order to solve this equation, one must express the size of the angle *θ* in radians. A radian is a unit for measuring the size of the angle in which 1 radian equals 180/*π*. (The symbol *π* [pi] is the ratio of the circumference of a circle to its diameter, and it is always the same, 3.141592+, no matter the size of the circle.)

## Applications

The use of trigonometry has expanded beyond merely solving problems dealing with right triangles. Some of the most important applications today deal with the periodic nature of trigonometric functions. For example, the times of sunsets, sunrises, and comet appearances can all be calculated by using trigonometric functions. Such functions also can be used to describe seasonal temperature changes, the movement of waves in the ocean, and even the quality of a musical sound.

[*See also* **Function; Pythagorean theorem** ]

# trigonometry

trig·o·nom·e·try / ˌtrigəˈnämitrē/ • n. the branch of mathematics dealing with the relations of the sides and angles of triangles and with the relevant functions of any angles.DERIVATIVES: trig·o·no·met·ric / -nəˈmetrik/ adj.trig·o·no·met·ri·cal / -nəˈmetrikəl/ adj.

# trigonometric function

**trigonometric function** Six ratios of the sides of a right-angled triangle containing a given acute angle – they are the sine, cosine, tangent, cotangent, secant, and cosecant of the angle. These functions can be extended to cover angles of any size by the use of a system of rectangular co-ordinates.

# trigonometry

# trigonometry

**trigonometry** Use of ratios of the sides of a right-angled triangle to calculate lengths and angles in geometrical figures. If three sides, or two sides and the included angle, or one side and two angles of a triangle are known, then all the other sides and angles may be found.

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