548 CHAPTER 8 Applications of Trigonometric Functions 2 Use the Complementary Angle Theorem Two acute angles are called complementary if their sum is a right angle, or 90 .° Because the sum of the angles of any triangle is 180 ,° it follows that, for a right triangle, the sum of the acute angles is 90 ,° so the two acute angles in every right triangle are complementary. Refer now to Figure 6, which labels the angle opposite side b as B and the angle opposite side a as A. Notice that side b is adjacent to angle A and side a is adjacent to angle B. As a result, Figure 6 c a b Adjacent to A Opposite B Adjacent to B Opposite A B A Figure 5(a) 0° 90° 28 24 Figure 5(b) Figure 7 Right triangle c a b B A The area A of the opening is A A area of the two congruent triangles area of the rectangle 16sin cos 16 sin 16 sin cos 1 θ θ θ θ θ θ ( ) ( ) = + = + = + (b) Figure 5(a) shows the graph of A A θ( ) = on a TI-84 Plus CE. Using MAXIMUM, the angle θ that makes A largest is 60 .° Figure 5(b) shows the maximum using Desmos. B b c A B a c A B b a A B c b A B c a A B a b A • sin cos •cos sin • tan cot •csc sec • sec csc •cot tan = = = = = = = = = = = = (2) THEOREM Complementary Angle Theorem Cofunctions of complementary angles are equal. Because of these relationships, the functions sine and cosine, tangent and cotangent, and secant and cosecant are called cofunctions of each other. The identities (2) may be expressed in words as follows: Examples of this theorem are given next: Complementary angles Complementary angles Complementary angles sin 30 cos60 tan 40 cot50 sec 80 csc10 ° = ° ° = ° ° = ° Cofunctions Cofunctions Cofunctions Using the Complementary Angle Theorem (a) sin62 cos 90 62 cos28 ( ) ° = °− ° = ° (b) tan 12 cot 2 12 cot 5 12 π π π π ( ) = − = (c) sin 40 sin 50 sin 40 cos 40 1 2 2 2 2 ° + ° = ° + ° = ↑ sin50 cos40 ° = ° EXAMPLE 3 Now Work PROBLEM 1 9 3 Solve Right Triangles In the discussion that follows, a right triangle is always labeled so that side a is opposite angle A, side b is opposite angle B, and side c is the hypotenuse, as shown in Figure 7. To solve a right triangle means to find the lengths of its sides and the measurements of its angles. We express the lengths of the sides rounded to two decimal places and angle measures in degrees rounded to one decimal place. (Be sure that your calculator is in degree mode.)
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