91Ó°ÊÓ

Find the angle of elevation (in degrees) for the given Mach number. Remember that an angle of elevation must be between \(0^{\circ}\) and \(90^{\circ}\). $$m=1.1$$

Prove the subtraction identity for sine: $$ \sin (x-y)=\sin x \cos y-\cos x \sin y $$ I Hint: Use the first cofunction identity* $$ \sin (x-y)=\cos \left[\frac{\pi}{2}-(x-y)\right]=\cos \left[\left(\frac{\pi}{2}-x\right)+y\right] $$ and the addition identity for cosine. \(]\)

Find the exact functional value without using a calculator. $$\tan ^{-1}[\tan (-4 \pi / 3)]$$

Write each expression as a sum or difference. $$\sin 17 x \sin (-3 x)$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\sin ^{2} x-\tan ^{2} x=-\sin ^{2} x \tan ^{2} x$$

Prove the addition identity for sine: $$ \sin (x+y)=\sin x \cos y+\cos x \sin y $$ [Hint: You may assume Exercise \(36 .\) Use the same method by which the addition identity for cosine was obtained from the subtraction identity for cosine in the text.]

Find the exact functional value without using a calculator. $$\left.\sin \left[\cos ^{-1}(3 / 5)\right] \text { (See Example } 11 .\right)$$

Write each expression as a sum or difference. $$\cos 13 x \cos (-5 x)$$

Find the exact functional value without using a calculator. $$\tan \left[\sin ^{-1}(3 / 5)\right]$$

Prove the addition and subtraction identities for the tangent function (page 526 ). [ Hint: $$ \tan (x+y)=\frac{\sin (x+y)}{\cos (x+y)} $$ Use the addition identities on the numerator and denominator; then divide both numerator and denominator by \(\cos x \cos y \text { and simplify. }]\)