Chapter 7

(a) Express the rule of the function \(f(x)=\cos ^{3} x\) in terms of constants and first powers of the cosine function as in Example 4. (b) Do the same for \(f(x)=\cos ^{4} x\)

Write the expression as an algebraic expression in \(v\). $$\sin \left(\tan ^{-1} v\right)$$

$$\text { Prove that } \cot (x+y)=\frac{\cot x \cot y-1}{\cot x+\cot y}$$.

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

Simplify the given expression. $$\frac{\sin 2 x}{2 \sin x}$$

Write the expression as an algebraic expression in \(v\). $$\cos \left(\tan ^{-1} v\right)$$

$$\text { Prove the identity.}$$ $$\sin (x-\pi)=-\sin x$$

Simplify the given expression. $$1-2 \sin ^{2}\left(\frac{x}{2}\right)$$

Write the expression as an algebraic expression in \(v\). $$\sin \left(2 \sin ^{-1} v\right)$$

$$\text { Prove the identity.}$$ $$\cos (x-\pi)=-\cos x$$

Simplify the given expression. $$2 \cos 2 y \sin 2 y(\text { Think } !)$$

Write the expression as an algebraic expression in \(v\). $$\sin \left(2 \cos ^{-1} v\right)$$

$$\text { Prove the identity.}$$ $$\cos (\pi-x)=-\cos x$$

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

Simplify the given expression. $$\cos ^{2}\left(\frac{x}{2}\right)-\sin ^{2}\left(\frac{x}{2}\right)$$

$$\text { Prove the identity.}$$ $$\tan (\pi-x)=-\tan x$$

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

Simplify the given expression. $$(\sin x+\cos x)^{2}-\sin 2 x$$

$$\text { Prove the identity.}$$ $$\sin (x+\pi)=-\sin x$$

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

$$\text { Prove the identity.}$$ $$\cos (x+\pi)=-\cos x$$

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

Prove the given sum to product identity. $$\sin x-\sin y=2 \cos \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)$$

Graph the function. $$f(x)=\cos ^{-1}(x+1)$$

$$\text { Prove the identity.}$$ $$\tan (x+\pi)=\tan x$$

Prove the given sum to product identity. $$\cos x+\cos y=2 \cos \left(\frac{x+y}{2}\right) \cos \left(\frac{x-y}{2}\right)$$

Graph the function. $$g(x)=\tan ^{-1} x+\pi$$

$$\text { Prove the identity.}$$ $$\sin x \cos y=\frac{1}{2}[\sin (x+y)+\sin (x-y)]$$

Prove the given sum to product identity. $$\cos x-\cos y=-2 \sin \left(\frac{x+y}{2}\right) \sin \left(\frac{x-y}{2}\right)$$

Graph the function. $$h(x)=\sin ^{-1}(\sin x)$$

$$\text { Prove the identity.}$$ $$\sin x \sin y=\frac{1}{2}[\cos (x-y)-\cos (x+y)]$$

Graph the function. $$k(x)=\sin \left(\sin ^{-1} x\right)$$

$$\text { Prove the identity.}$$ $$\cos x \sin y=\frac{1}{2}[\sin (x+y)-\sin (x-y)]$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$3 \sin ^{2} x-8 \sin x-3=0$$

In an alternating current circuit, the voltage is given by the formula $$V=V_{\max } \cdot \sin (2 \pi f t+\phi)$$ where \(V_{\max }\) is the maximum voltage, \(f\) is the frequency (in cycles per second), \(t\) is the time in seconds, and \(\phi\) is the phase angle. (a) If the phase angle is \(0,\) solve the voltage equation for \(t\) (b) If \(\phi=0, V_{\max }=20, V=8.5,\) and \(f=120,\) find the smallest positive value of \(t\)

$$\text { Prove the identity.}$$ $$\cos (x+y) \cos (x-y)=\cos ^{2} x \cos ^{2} y-\sin ^{2} x \sin ^{2} y$$

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

Determine graphically whether the equa. tion could possibly be an identity. If it could, prove that it is. $$\cos 8 x=\cos ^{2} 4 x-\sin ^{2} 4 x$$

Calculus can be used to show that the area \(A\) between the \(x\) axis and the graph of \(y=\frac{1}{x^{2}+1}\) from \(x=a\) to \(x=b\) is given by \(A=\tan ^{-1} b-\tan ^{-1} a\) Find the area \(A\) when (a) \(a=0\) and \(b=1\) (b) \(a=-1\) and \(b=2\) (c) \(a=-2.5\) and \(b=-.5\) (GRAPH CANNOT COPY)

$$\text { Prove the identity.}$$ $$\sin (x+y) \sin (x-y)=\sin ^{2} x \cos ^{2} y-\cos ^{2} x \sin ^{2} y$$

State whether or not the equation is an identity. If it is an identity, prove it. $$\frac{\cot x}{\csc x-1}=\frac{\csc x+1}{\cot x}$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$2 \tan ^{2} x+7 \tan x+5=0$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$3 \sin ^{2} x+2 \sin x=5$$

Suppose that another model plane is flying while attached to the ground by a 100 foot long wire that is always kept taut. Let \(h\) denote the height of the plane above the ground and \(\theta\) the radian measure of the angle the wire makes with the ground. (The figure for Exercise 65 is the case when \(x=\) \(100 \text { and } h=40 .)\) (a) Express \(\theta\) as a function of the height \(h\) (b) What is \(\theta\) when the plane is 55 feet above the ground? (c) When \(\theta=1\) radian, how high is the plane?

A rocket is fired straight up. The line of sight from an observer 4 miles away makes an angle of \(t\) radians with the horizontal. (a) Express \(t\) as a function of the height \(h\) of the rocket. (b) Find \(t\) when the rocket is .25 mile, 1 mile, and 2 miles high respectively. (c) When \(t=.4\) radian, how high is the rocket? (GRAPH CANNOT COPY)

Half of an identity is given. Graph this half in a viewing window with \(-2 \pi \leq x \leq 2 \pi\) and make a conjecture as to what the right side of the identity is. Then prove your conjecture. $$\cos ^{3} x\left(1-\tan ^{4} x+\sec ^{4} x\right)=?$$

Use factoring, the quadratic formula, or identities to solve the equation. Find all solutions in the interval \([0,2 \pi)\). $$\cos x \csc x=2 \cos x$$

Prove the identity. $$\frac{1-\sin x}{\sec x}=\frac{\cos ^{3} x}{1+\sin x}$$

Prove the identity. $$\frac{\sin x}{1-\cot x}+\frac{\cos x}{1-\tan x}=\cos x+\sin x$$

Prove the identity. $$\frac{\cos x}{1-\sin x}=\sec x+\tan x$$