550 CHAPTER 5 Trigonometric Functions NOTE The cosecant, secant, and cotangent ratios are reciprocals of the sine, cosine, and tangent values, respectively, so in Example 1 we have csc A = 25 7 sec A = 25 24 cot A = 24 7 csc B = 25 24 sec B = 25 7 and cot B = 7 24 . The cofunction identities state the following. Cofunction values of complementary angles are equal. EXAMPLE 2 Writing Functions inTerms of Cofunctions Write each function in terms of its cofunction. (a) cos 52° (b) tan 71° (c) sec 24° SOLUTION (a) Cofunctions Complementary angles cos 52° = sin190° - 52°2 = sin 38° cos A = sin190° - A2 (b) tan 71° = cot190° - 71°2 = cot 19° (c) sec 24° = csc 66° S Now Try Exercises 27 and 29. Cofunctions Figure 29 shows a right triangle with acute angles A and B and a right angle at C. The length of the side opposite angle A is a, and the length of the side opposite angle B is b. The length of the hypotenuse is c. By the preceding definitions, sin A = a c . Also, cos B = a c . Thus, we have the following. sin A = a c =cos B Similarly, tan A = a b =cot B and sec A = c b =csc B. In any right triangle, the sum of the two acute angles is 90°, so they are complementary. In Figure 29, A and B are thus complementary, and we have established that sin A = cos B. This can also be written as follows. sin A = cos190° - A2 B = 90° - A This is an example of a more general relationship between cofunction pairs. sine, cosine tangent, cotangent secant, cosecant A C a c b B Whenever we use A, B, and C to name angles in a right triangle, C will be the right angle. Figure 29 Cofunction pairs (+)+* Cofunction Identities For any acute angle A, the following hold true. sin A =cos190° −A2 sec A =csc190° −A2 tan A =cot190° −A2 cos A =sin190° −A2 csc A =sec190° −A2 cot A =tan190° −A2
RkJQdWJsaXNoZXIy NjM5ODQ=