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: a² = b² + c² − 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 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 2π 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 [Gr.,=measurement of triangles], a specialized area of geometry concerned with the properties of and relations among the parts of a triangle. Spherical trigonometry is concerned with the study of triangles on the surface of a sphere rather than in the plane; it is of considerable importance in surveying, navigation, and astronomy.

The Basic Trigonometric Functions

Trigomometry originated as the study of certain mathematical relations originally defined in terms of the angles and sides of a right triangle, i.e., one containing a right angle (90°). Six basic relations, or trigonometric functions, are defined. If A, B, and C are the measures of the angles of a right triangle ( C =90°) and a, b, and c are the lengths of the respective sides opposite these angles, then the six functions are expressed for one of the acute angles, say A, as various ratios of the opposite side ( a ), the adjacent side ( b ), and the hypotenuse ( c ), as set out in the table. Although the actual lengths of the sides of a right triangle may have any values, the ratios of the lengths will be the same for all similar right triangles, large or small; these ratios depend only on the angles and not on the actual lengths. The functions occur in pairs—sine and cosine, tangent and cotangent, secant and cosecant—called cofunctions. In equations they are usually represented as sin, cos, tan, cot, sec, and csc. Since in ordinary (Euclidean) plane geometry the sum of the angles of a triangle is 180°, angles A and B must add up to 90° and therefore are complementary angles. From the definitions of the functions, it may be seen that sin B =cos A, cos B =sin A, tan B =cot A, and sec B =csc A ; in general, the function of an angle is equal to the cofunction of its complement. Since the hypotenuse ( c ), is always the longest side of a right triangle, the values of the sine and cosine are always between zero and one, the values of the secant and cosecant are always equal to or greater than one, and the values of the tangent and cotangent are unbounded, increasing from zero without limit.

For certain special right triangles the values of the functions may be calculated easily; e.g., in a right triangle whose acute angles are 30° and 60° the sides are in the ratio 1 : 3  : 2, so that sin 30°=cos 60°=1/2, cos 30°=sin 60°= 3 /2, tan 30°=cot 60°=1/ 3 , cot 30°=tan 60°= 3 , sec 30°=csc 60°=2/ 3 , and csc 30°=sec 60°=2. For other angles, the values of the trigonometric functions are usually found from a set of tables or a scientific calculator. For the limiting values of 0° and 90°, the length of one side of the triangle approaches zero while the other approaches that of the hypotenuse, resulting in the values sin 0°=cos 90°=0, cos 0°=sin 90°=1, tan 0°=cot 90°=0, and sec 0°=csc 90°=1; since division by zero is undefined, cot 0°, tan 90°, csc 0°, and sec 90° are all undefined, having infinitely large values.

A general triangle, not necessarily containing a right angle, can also be analyzed by means of trigonometry, and various relationships are found to exist between the sides and angles of the general triangle. For example, in any plane triangle a /sin A = b /sin B = c /sin C. This relationship is known as the Law of Sines. The related Law of Cosines holds that a² = b² + c² -2 bc cos A and the Law of Tangents holds that ( a - b )/( a + b )=[tan 1/2 ( A - B )]/[tan 1/2 ( A + B )]. Each of the trigonometric functions can be represented by an infinite series .

Extension of the Trigonometric Functions

The notion of the trigonometric functions can be extended beyond 90° by defining the functions with respect to Cartesian coordinates . Let r be a line of unit length from the origin to the point P ( x,y ), and let θ be the angle r makes with the positive x -axis. The six functions become sin θ = y / r = y, cos θ= x / r = x, tan θ= y / x, cot θ= x / y, sec θ= r / x =1/ x, and csc θ= r / y =1/ y. As θ increases beyond 90°, the point P crosses the y -axis and x becomes negative; in quadrant II the functions are negative except for sin θ and csc θ. Beyond θ=180°, P is in quadrant III, y is also negative, and only tan θ and cot θ are positive, while beyond θ=270° P moves into quadrant IV, x becomes positive again, and cos θ and sec θ are positive. Since the positions of r for angles of 360° or more coincide with those already taken by r as θ increased from 0°, the values of the functions repeat those taken between 0° and 360° for angles greater than 360°, repeating again after 720°, and so on. This repeating, or periodic, nature of the trigonometric functions leads to important applications in the study of such periodic phenomena as light and electricity.

Trigonometric Functions

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 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 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.

trigonometry branch of mathematics dealing with the measurements of triangles. XVII. — modL. trigōnometria, f. Gr. trígōnon triangle; see TRI-, -GON, -METRY.

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