549 5.3 Trigonometric Function Values and Angle Measures 5.3 Trigonometric Function Values and Angle Measures Right-Triangle-Based Definitions of the Trigonometric Functions Angles in standard position can be used to define the trigonometric functions. There is also another way to approach them: as ratios of the lengths of the sides of right triangles. Figure 27 shows an acute angle A in standard position. The definitions of the trigonometric function values of angle A require x, y, and r. As drawn in Figure 27, x and y are the lengths of the two legs of the right triangle ABC, and r is the length of the hypotenuse. The side of length y is the side opposite angle A, and the side of length x is the side adjacent to angle A. We use the lengths of these sides to replace x and y in the definitions of the trigonometric functions, and the length of the hypotenuse to replace r, to obtain the following righttriangle-based definitions. In the definitions, we use the standard abbreviations for the sine, cosine, tangent, cosecant, secant, and cotangent functions. ■ Right-Triangle-Based Definitions of the Trigonometric Functions ■ Cofunctions ■ Trigonometric Function Values of Special Angles ■ Reference Angles ■ Special Angles as Reference Angles ■ Determination of Angle Measures with Special Reference Angles ■ Calculator Approximations of Trigonometric Function Values ■ Calculator Approximations of Angle Measures ■ An Application A x y (x, y) C x y r B Figure 27 NOTE We will sometimes shorten wording like “side opposite A” to just “side opposite” when the meaning is obvious. A B 7 24 25 C Figure 28 sin A = side opposite hypotenuse = 7 25 cos A = side adjacent hypotenuse = 24 25 tan A = side opposite side adjacent = 7 24 The length of the side opposite angle B is 24, and the length of the side adjacent to angle B is 7. sin B = 24 25 cos B = 7 25 tan B = 24 7 Use the right-triangle-based definitions of the trigonometric functions. S Now Try Exercise 11. Right-Triangle-Based Definitions of Trigonometric Functions Let A represent any acute angle in standard position. sin A = y r = side opposite A hypotenuse csc A = r y = hypotenuse side opposite A cos A = x r = side adjacent to A hypotenuse sec A = r x = hypotenuse side adjacent to A tan A = y x = side opposite A side adjacent to A cot A = x y = side adjacent to A side opposite A EXAMPLE 1 FindingTrigonometric Function Values of an Acute Angle Find the sine, cosine, and tangent values for angles A and B in the right triangle in Figure 28. SOLUTION The length of the side opposite angle A is 7, the length of the side adjacent to angle A is 24, and the length of the hypotenuse is 25.
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