Trigonometry

views updated

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.