Chapter 20

Evaluate each of the following expressions exactly. Do not give numerical approximations. (a) \(\sin ^{-1}(1)\) (b) \(\tan ^{-1}(1)\) (c) \(\sin ^{-1}(-1)\) (d) \(\cos ^{-1}(-1)\) (e) \(\sin ^{-1}(-0.5)\) (f) \(\cos ^{-1}(-0.5)\) (g) \(\tan ^{-1}(\sqrt{3})\) (h) \(\left[\cos ^{-1}(-1)\right]^{2}+\left[\tan ^{-1}(-1)\right]^{2}\)

Find the following exactly. (a) \(\sin (\pi / 6)\) (b) \(\sin (\pi / 4)\) (c) \(\sin (\pi / 3)\) (d) \(\sin (-\pi / 3)\) (e) \(\cos (\pi / 3)\) (f) \(\cos (-\pi / 3)\) (g) \(\tan (\pi / 4)\) (h) \(\tan (3 \pi / 4)\) (i) \(\tan (5 \pi / 3)\) (j) \(\sin (-7 \pi / 6)\)

True or false: If the equation is not always true, give a counterexample. (a) \(\sin (A-B)=\sin (A)-\sin (B)\) (b) \(\cos (A+B)=\cos A+\cos B\)

Find all \(x\) between 0 and \(2 \pi\) such that (a) \(4 \cos ^{2} x=3\). (b) \(2 \sin ^{2} x-\sin x-1=0 . \quad\) (Hint: this is a quadratic in \(\sin x .\) ) (c) \(\sin x=2 / 3\). (Give exact answers, as well as numerical approximations.)

(a) For what values of \(x\) is \(\tan x=\sqrt{3}\) ? (b) For what values of \(x\) is \(\tan (x)=-\sqrt{3}\) ?

$$ |\overrightarrow{\mathbf{u}}|=5,|\overrightarrow{\mathbf{v}}|=7 \text { , and the angle between } \overrightarrow{\mathbf{u}} \text { and } \overrightarrow{\mathbf{v}} \text { is } \frac{\pi}{6} \text { . } $$

Compute the following exactly. Do not use calculator approximations. (a) \(\cos \left(\frac{\pi}{12}\right)=\cos \left(\frac{\pi}{4}-\frac{\pi}{6}\right)\) (b) \(\sin \left(\frac{-\pi}{12}\right)=\sin \left(\frac{\pi}{6}-\frac{\pi}{4}\right)\) (c) \(\tan \left(\frac{\pi}{12}\right)\)

Find all solutions. (a) \(\sin ^{2} x=0.25\). (b) \(\cos ^{2} x+2 \cos x+1=0\). (Hint: Let \(u=\cos x\) and first find \(u\).) (c) \(\cos ^{2} x+4 \cos x+3=0\)

The function \(f(x)=\tan x\) has an inverse function when its domain is restricted to \((-\pi / 2, \pi / 2)\) (a) Graph \(y=\tan ^{-1}(x)\). Where is the derivative positive? Negative? (b) Is \(\tan ^{-1}(x)\) an even function, an odd function, or neither? (c) Is the derivative of \(\tan ^{-1}(x)\) even, odd, or neither?

Find the exact values of the following. (a) \(\sin (2 \pi / 3)\) (b) \(\cos (5 \pi / 4)\) (c) \(\tan (7 \pi / 4)\) (d) \(\sec (\pi / 6)\) (e) \(\cot (-\pi / 6)\) (f) \(\csc (4 \pi / 3)\) (g) \(\sin (801 \pi)\) (h) \(\cos (39 \pi / 4)\)

$$ |\overrightarrow{\mathbf{u}}|=5,|\overrightarrow{\mathbf{v}}|=7 \text { , and the angle between } \overrightarrow{\mathbf{u}} \text { and } \overrightarrow{\mathbf{v}} \text { is } \frac{\pi}{3} \text { . } $$

Using the addition formulas and what you know about even and odd trigonometric functions, find expressions for \(\sin (x-2 y)\) and \(\cos (x-2 y)\) in terms of \(\cos x, \cos y, \sin x\) and \(\sin y\).

Find all solutions to the following equations: (a) \(\sec ^{2} x=2\). (b) \(\cos ^{2} x=0.2 \cos x\). Why can't you cancel the \(\cos x\) from both sides of the equation? (c) \(\sin ^{2} x=3 \cos x+1\).

A 25 -foot ladder is leaning against a straight wall. If the base of the ladder is 7 feet from the wall, what angle is the ladder making with the ground and how high up the wall does it go?

(a) Let \(h(x)=\sin \left(\sin ^{-1}(x)\right)\). What is the domain of \(h\) ? Is \(h(x)=x\) for every \(x\) in its domain? If not, for what \(x\) is \(\sin \left(\sin ^{-1}(x)\right) \neq x\) ? b) Let \(j(x)=\sin ^{-1}(\sin (x))\). What is the domain of \(j\) ? For which values of \(x\) is \(j(x)=x ?\) For which values of \(x\) is \(j(x) \neq x ?(\) Caution: There are infinitely many values of \(x\) for which \(j(x) \neq x .\) Be sure to identify them all.)

(a) Find an acute angle \(x\) such that \(\sin x=0.5\). (b) Find all \(x\) between 0 and \(2 \pi\) such that \(\sin x=0.5\). (c) Find an acute angle \(x\) such that \(\cos x=0.5\). (d) Find all \(x \in[0,2 \pi]\) such that \(\cos (2 x)=0.5\).

Match with each of the expressions on the right with the equivalent expression on the left. (a) \(\cos ^{2} x\) (i) \(1-\cos ^{2} x\) (b) \(\sin ^{2} x\) (ii) \(\quad 1-\sin ^{2} x\) (c) \(\tan ^{2} x\) (iii) \(\frac{1}{2}-\frac{1}{2} \cos 2 x\) (d) \(\sin x\) (iv) \(-1+\sec ^{2} x\) (e) \(\cos x\) (v) \(\quad 1-\sin ^{2}(x+\pi / 2)\) (f) \(-\cos x\) (vi) \(\sin (x-\pi / 2)\) (vii) \(\frac{1}{2}+\frac{1}{2} \cos 2 x\) (viii) \(\cos (x-\pi / 2)\)

You're in an apartment looking across a 25 -foot boulevard at a building across the way. The angle of depression to the foot of the building is \(15^{\circ}\) and the angle of elevation to the top of the building is \(50^{\circ} .\) How tall is the building?

On the same set of axes sketch the graphs of \(f(x)\) and \(f^{-1}(x)\). (a) \(f(x)=\sin x\), where \(x \in[-\pi / 2, \pi / 2]\). (b) \(f(x)=\cos x\), where \(x \in[0, \pi]\).

Use the addition formulas for \(\sin x\) and \(\cos x\) to derive the addition and subtraction formulas for \(\tan x\). $$ \begin{array}{l} \tan (A+B)=\frac{\tan A+\tan B}{1-\tan A \tan B} \\ \tan (A-B)=\frac{\tan A-\tan B}{1+\tan A \tan B} \end{array} $$

After a hurricane a tree is left standing but makes an angle of \(10^{\circ}\) with its former upright position. Suppose the tree is tilting away from the sun and casting a shadow 25 feet long. If the angle of elevation of the sun is \(30^{\circ}\), how long is the tree?

You are standing on an overpass, 35 feet above street level, waving to a friend who is at the window of a high-rise dormitory. The angle of depression to the bottom of the dormitory (on street level) is \(15^{\circ}\). The angle of elevation to your friend's window is \(45^{\circ}\). What is your friend's elevation?

Find the angle between \(\pi / 2\) and \(\pi\) whose sine is (a) \(0.5\). (b) 0.2. (Give an exact answer and then a numerical approximation.)

Fill in the following table. $$ \begin{array}{l} \theta \text { in degrees } \theta \text { in radians } \sin \theta \quad \cos \theta \quad \tan \theta \\ \hline 30^{\circ} \\ \hline 45^{\circ} \\ \hline 60^{\circ} \\ \hline \end{array} $$

Suppose \(\overrightarrow{\mathrm{v}}\) is a vector of length 8 and the component of \(\overrightarrow{\mathbf{v}}\) in the direction of \(\overrightarrow{\mathrm{u}}\) is \(4 .\) (a) Can the angle between \(\overrightarrow{\mathrm{u}}\) and \(\overrightarrow{\mathrm{v}}\) be determined? If so, what is it? (b) Can the direction of \(\overrightarrow{\mathrm{u}}\) be determined? If so, what is it? (c) Can the length of \(\overrightarrow{\mathrm{u}}\) be determined? If so, what is it?

Use the addition formula for \(\tan (A+B)\) to show that $$ \tan 2 x=\frac{2 \tan x}{1-\tan ^{2} x} $$

Use a graph to check that you have found all solutions in this interval. (Check \(f(x)=0.5\) on \([0,2 \pi]\) by graphing \(y=f(x)\) and \(y=0.5\) on \([0,2 \pi]\) and looking for points of intersection or by graphing \(y=f(x)-0.5\) on \([0,2 \pi]\) and looking for zeros. \()\) $$ \cos (2 x)=1 $$

We are standing on flat ground in Monument Valley trying to estimate the height of the edifices. We have surveying equipment and take all of our measurements from a height of 5 feet. We find the angle of elevation to the top of one structure is \(23^{\circ} .\) We move 500 feet closer to the structure and find that the angle of elevation is now \(29^{\circ} .\) How tall is the structure?

Simplify the following. (a) \(\sin \left(\arctan \frac{3}{4}\right)\) (b) \(\tan \left(\cos ^{-1}(0.5)\right)\) (c) \(\cos \left(\sin ^{-1} x\right), \quad x<0\) (d) \(\tan \left(\sin ^{-1}\left(\frac{w}{r}\right)\right), \quad w, r>0\)

Find exact values for each of the following. (No calculator-or use it only to check your answers.) (a) \(\cos (\pi / 4)\) (b) \(\cos (5 \pi / 4)\) (c) \(\cos (-3 \pi / 4)\) (d) \(\sin (5 \pi / 6)\) (e) \(\sin (-13 \pi / 6)\) (f) \(\cos (-2 \pi / 3)\)

Solve. (a) \(\cos ^{2} x-\cos 2 x=0 \quad x \in(-\infty, \infty)\) (b) \(\sin x \cos x=\sqrt{3} \quad x \in[0,2 \pi]\) (c) \(\sin ^{2} x-\cos 2 x=0 \quad x \in[0,2 \pi]\) (d) \(-\cos ^{2} x+\frac{1}{2} \sin x+1=0 \quad x \in[0,2 \pi]\)

Use a graph to check that you have found all solutions in this interval. (Check \(f(x)=0.5\) on \([0,2 \pi]\) by graphing \(y=f(x)\) and \(y=0.5\) on \([0,2 \pi]\) and looking for points of intersection or by graphing \(y=f(x)-0.5\) on \([0,2 \pi]\) and looking for zeros. \()\) $$ 2 \sin ^{2}(2 x)=1 $$

Graph each of the following. Be sure to display at least one full period of the function. For parts (a), (b), and (e), use what you know about graphing \(\frac{1}{f(x)}\) when given the graph of \(f(x)\). (a) \(y=3 \sec x\) (b) \(y=-\cot x\) (c) \(y=\tan \left(\frac{x}{2}\right)\) (d) \(y=-3 \tan x\) (e) \(y=2 \csc x\)

Let \(f(x)=\cos x\) and \(g(x)=\arctan x .\) Find the following, where \(a\) and \(b\) are positive constants. Your answers should be exact and as simple as possible. (a) \(g(f(\pi))\) (b) \(f\left(g\left(\frac{-a}{h}\right)\right)\)

A plane is traveling 300 miles per hour. There is a 50 -mph wind. The angle between he velocity vector of the wind and the velocity vector of the plane is \(110^{\circ}\). (a) What is the component of the direction of the plane's motion? (b) In the absence of the wind but all else remaining the same, how fast would the plane be traveling?

\text { Write a formula for } \cos 3 x \text { entirely in terms of sums and powers of } \cos x \text { . }

In several ancient civilizations trigonometry was a highly developed field. This can in part be attributed to ancient astronomers. How could an ancient astronomer who could measure angles of elevation use trigonometry to estimate the height of the moon?

In Problems 9 through 11, simplify the expressions given that \(x \in\left[0, \frac{\pi}{2}\right]\). $$ \text { (a) } \sin ^{-1}(\sin x) $$

Some kids are sitting on their stoop wondering about the height of a tall street post. They estimate that the street post is casting a shadow 15 feet in length and that the angle of elevation of the sun (from the ground) is about \(60^{\circ}\). Estimate the height of the street post.

Suppose an object is launched from a height of 64 feet with an initial velocity of 96 feet per second at an angle of \(\pi / 6\) radians. Assume that the only force acting on the object is the force of gravity, which results in a downward acceleration of \(32 \mathrm{ft} / \mathrm{sec} .\) (a) Find the vertical position of the object at time \(t\). (b) When will the object hit the ground? (c) How far has the object traveled horizontally when it hits the ground? (In other words, what is the horizontal component of its displacement vector?) (d) When the object hits the ground, how far is it from where it was launched? (In other words, what is the length of its displacement vector?)

Use the power-reducing formulas for \(\sin ^{2} x\) and \(\cos ^{2} x\) to show that $$ \tan ^{2} x=\frac{1-\cos 2 x}{1+\cos 2 x} $$

Use a graph to check that you have found all solutions in this interval. (Check \(f(x)=0.5\) on \([0,2 \pi]\) by graphing \(y=f(x)\) and \(y=0.5\) on \([0,2 \pi]\) and looking for points of intersection or by graphing \(y=f(x)-0.5\) on \([0,2 \pi]\) and looking for zeros. \()\) $$ \cos ^{2} x+4 \sin x=4 $$

Peter is measuring the height of a church steeple. He stands on level ground 500 feet from the base of the church and determines that the angle of elevation from the ground to the base of the steeple is \(23^{\circ}\). From the same spot he measures the angle of elevation to the highest point of the steeple and finds it is \(29^{\circ}\). (a) How high is the church, from the base of the church at ground level to the tip of the steeple? Give an exact answer and then give a numerical approximation. (b) How high is the steeple? Give an exact answer and then give a numerical approximation.

For Problems 10 through 14, rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\sin \left(-48^{\circ}\right)\) (b) \(\cos \left(-48^{\circ}\right)\)

Force \(A\) has a horizontal component of 3 pounds and a vertical component of 4 pounds. Force \(B\) has a horizontal component of 5 pounds and a vertical component of 12 pounds. (a) What is the strength of force \(A\) ? What angle does this force vector make with the horizontal? (Give a numerical approximation in degrees.) (b) What is the strength of force \(B\) ? What angle does this force vector make with the horizontal? (Give a numerical approximation in degrees.) (c) What is the component of force \(A\) in the direction of force \(B\) ?

When designing the one-third-of-a-mile-long Georgia World Congress Center, the building that housed nearly one-fifth of the events of the 1996 Olympics, engineers had to take into account the curvature of the earth (Sports Illustrated, August 5,1996 ). Assuming a constant curvature of the earth, how many feet would it curve in one-third of a mile? In other words, assume a cross-section of the earth is a perfect circle and draw a tangent line to the curve of this circle at one end of the building. How far away would the tangent line be from the circle itself at the other end of the building? (Use 3960 miles as the radius of the earth.)

Use a graph to check that you have found all solutions in this interval. (Check \(f(x)=0.5\) on \([0,2 \pi]\) by graphing \(y=f(x)\) and \(y=0.5\) on \([0,2 \pi]\) and looking for points of intersection or by graphing \(y=f(x)-0.5\) on \([0,2 \pi]\) and looking for zeros. \()\) $$ \cos (3 x)=0.5 $$

Simplify the expressions given that \(x \in\left[0, \frac{\pi}{2}\right]\). (a) \(\tan ^{-1}(\tan (x))\) (b) \(\tan ^{-1}(\tan (-x))\)

Rewrite each of the following expressions in terms of a positive acute angle. This positive acute angle is sometimes referred to as a reference angle. (a) \(\sin 92^{\circ}\) (b) \(\sin \left(-92^{\circ}\right)\)

There's a 10 -mile-per-hour wind, but a bicyclist calculates that its component in his direction of motion is only 9 miles per hour. What is the angle between the velocity vector of the wind and that of the bicyclist?