Chapter 7: Analytic Trigonometry

In Problems 81–92, use a graphing utility to solve each equation. Express the solution(s) rounded to two decimal places.

$x+5\cos x=0$

Establish each identity.

$\frac{{\sin }^3\theta +{\cos }^3\theta }{\sin \theta +\cos \theta }=1-\sin \theta \cos \theta$

In Problems 79-90, find the exact value of each expression.

$\cos \left(2{\sin }^{-1}\frac{3}{5}\right)$

Find the exact value of each expression.

$\tan \left({\sin }^{-1}\frac{3}{5}+\frac{\mathrm{\pi }}{6}\right)$

A projectile fired into the first quadrant from

the origin of a coordinate system will pass through the point

(x, y) at time t according to the relationship $cot\theta =\frac{2x}{2y+g{t}^2}$,

where $\theta$= the angle of elevation of the launcher and g =

the acceleration due to gravity = 32.2$ft/se{c}^2$

. An artilleryman is firing at an enemy bunker located 2450 feet

up the side of a hill that is 6175 feet away. He fires a round,

and exactly 2.27 seconds later he scores a direct hit.

(a) What angle of elevation did he use?

(b) If the angle of elevation is also given by $sec\theta =\frac{v_{0}t}{x}$

x , where$v_0$is the muzzle velocity of the weapon, find the muzzle

velocity of the artillery piece he used.

Use a graphing utility to solve the equation on the interval $0\le x\le 2\mathrm{\pi }$. Approximate any solutions rounded to two decimal places.

$2\sin x+3\cos x=4x$

In Problems 79-90, find the exact value of each expression.

$\cos \left(2{\cos }^{-1}\frac{4}{5}\right)$

Using a graphing utility, graph $y=co{t}^{-1}x$.

Find the exact value of each expression.

$\tan \left(\frac{\mathrm{\pi }}{4}-{\cos }^{-1}\frac{3}{5}\right)$

Establish each identity.

$\frac{{\sin }^3\theta +{\cos }^3\theta }{1-2{\cos }^2\theta }=\frac{sec\theta -\sin \theta }{\tan \theta -1}$