Chapter 7

Given that \(u=\sin ^{-1}(-\sqrt{3} / 2),\) find the exact value of \(\cos u\) and \(\tan u\)

Simplify the given expression. $$\text { If } \sin x=\frac{1}{3} \text { and } 0

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\tan \theta=7.95$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\sin x=\sqrt{1-\cos ^{2} x}$$

Use the half-angle identities to evaluate the given expression exactly. $$\cos \frac{\pi}{24}$$

Given that \(u=\tan ^{-1}(4 / 3),\) find the exact value of \(\sin u\) and \(\sec u\)

Simplify the given expression. $$\text { If } \cos x=-\frac{1}{4} \text { and } \frac{\pi}{2}

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\tan \theta=69.4$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\cot x=\frac{\csc x}{\sec x}$$

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) under the given conditions. $$\cos x=.4 \quad\left(0

Simplify the given expression. $$\text { If } \cos x=-\frac{1}{5} \text { and } \pi

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\cos \theta=-.42$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{\sin (-x)}{\cos (-x)}=-\tan x$$

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) under the given conditions. $$\sin x=.6 \quad\left(\frac{\pi}{2}

Find the exact functional value without using a calculator. $$\cos ^{-1}(\sin \pi / 6)$$

Simplify the given expression. $$\text { If } \sin x=-\frac{3}{4} \text { and } \frac{3 \pi}{2}

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\cot \theta=-2.4$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\tan x=\sqrt{\sec ^{2} x-1}$$

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) under the given conditions. $$\sin x=-\frac{3}{5} \quad\left(\frac{3 \pi}{2}

Assume that \(\sin x=.8\) and \(\sin y=\sqrt{.75}\) and that \(x\) and y lie between 0 and \(\pi / 2\). Evaluate the given expressions. $$\sin (x+y)$$

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$2 \sin ^{2} \theta+3 \sin \theta+1=0$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\cot (-x)=-\cot x$$

Find the exact functional value without using a calculator. $$\tan ^{-1}(\cos \pi)$$

Assume that \(\sin x=.8\) and \(\sin y=\sqrt{.75}\) and that \(x\) and y lie between 0 and \(\pi / 2\). Evaluate the given expressions. $$\cos (x-y)$$

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$4 \cos ^{2} \theta+4 \cos \theta-3=0$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\sec (-x)=\sec x$$

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) under the given conditions. $$\tan x=\frac{1}{2} \quad\left(\pi

Find the exact functional value without using a calculator. $$\sin ^{-1}(\cos 7 \pi / 6)$$

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\tan ^{2} \theta-3=0$$

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) under the given conditions. $$\cot x=1 \quad\left(-\pi

Find the exact functional value without using a calculator. $$\cos ^{-1}(\tan 7 \pi / 4)$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\sec ^{4} x-\tan ^{4} x=1+2 \tan ^{2} x$$

Find the exact functional value without using a calculator. $$\left.\sin ^{-1}(\sin 2 \pi / 3) \text { (See Exercise } 19 .\right)$$

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

Find the exact functional value without using a calculator. $$\cos ^{-1}(\cos 5 \pi / 4)$$

Find all angles \(\theta\) with \(0^{\circ} \leq \theta<360^{\circ}\) that are solutions of the given equation. IHint: Put your calculator in degree mode and replace \(\pi\) by \(180^{\circ}\) in the solution algorithms for basic equations. \(]\) $$\sin ^{2} \theta-3 \sin \theta=10$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\sec ^{2} x+\csc ^{2} x=\sec ^{2} x \csc ^{2} x$$

Find the exact functional value without using a calculator. $$\cos ^{-1}[\cos (-\pi / 6)]$$

If \(f(x)=\cos x\) and \(h\) is a fixed nonzero number, prove that: \(\frac{f(x+h)-f(x)}{h}=\cos x\left(\frac{\cos h-1}{h}\right)-\sin x\left(\frac{\sin h}{h}\right)\).

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

State whether or not the equation is an identity. If it is an identity, prove it. $$\sin ^{2} x(\cot x+1)^{2}=\cos ^{2} x(\tan x+1)^{2}$$

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

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. \(]\)

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

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

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.]

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

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 \(\cos x \cos y \text { and simplify. }]\)