Chapter 6

Factor the given expression. $$\cos ^{2} t-\cos t-2$$

Find tan \(t,\) where the terminal side of an angle of t radians lies on the given line. $$y=32 x$$

In Exercises \(1-10,\) use the definition (not a calculator) to find the function value. $$\tan (-13 \pi)$$

Give the rule of a periodic function with the given numbers as amplitude, period, and phase shift (in this order) $$3 / 4,2,0$$

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(-1 / 5,1)$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\sin t ; \quad g(t)=\sin t+3$$

In Exercises \(11-14\), assume that the terminal side of an angle of \(t\) radians passes through the given point on the unit circle. Find \(\sin t, \cos t,\) tan \(t\) $$(-2 / \sqrt{5}, 1 / \sqrt{5})$$

Find the radian measure of four angles in standard position that are coterminal with the angle in stan- dard position whose measure is given. $$\pi / 4$$

Give the rule of a periodic function with the given numbers as amplitude, period, and phase shift (in this order) $$4 / 5,3,1$$

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(4 / 5,-3 / 5)$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\cos t ; \quad g(t)=\cos t-2$$

Factor the given expression. $$\tan t \cos t+\cos ^{2} t$$

In Exercises \(11-14\), assume that the terminal side of an angle of \(t\) radians passes through the given point on the unit circle. Find \(\sin t, \cos t,\) tan \(t\) $$(1 / \sqrt{10},-3 / \sqrt{10})$$

Find the radian measure of four angles in standard position that are coterminal with the angle in stan- dard position whose measure is given. $$7 \pi / 5$$

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(\sqrt{2}, \sqrt{3})$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\cos t ; \quad g(t)=-\cos t$$

Factor the given expression. $$\cos ^{4} t+4 \cos ^{2} t-5$$

In Exercises \(11-14\), assume that the terminal side of an angle of \(t\) radians passes through the given point on the unit circle. Find \(\sin t, \cos t,\) tan \(t\) $$(-3 / 5,-4 / 5)$$

Find the radian measure of four angles in standard position that are coterminal with the angle in stan- dard position whose measure is given. $$-\pi / 6$$

Give the rule of a periodic function with the given numbers as amplitude, period, and phase shift (in this order) $$18,3,-6$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(x)=\sin t ; \quad g(t)=-3 \sin t$$

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(-2 \sqrt{3}, \sqrt{3})$$

In Exercises \(11-14\), assume that the terminal side of an angle of \(t\) radians passes through the given point on the unit circle. Find \(\sin t, \cos t,\) tan \(t\) $$(.6,-.8)$$

State the rule of a function of the form \(f(t)=A \sin b t\) or \(g(t)=A \cos b t\) whose graph appears to be identical to the given graph. (Check your book to see graph)

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\tan t ; \quad g(t)=\tan t+5$$

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(1+\sqrt{2}, 3)$$

Find the rules of the composite functions \(f \circ g\) and \(g \circ f\). $$f(t)=\cos t ; \quad g(t)=2 t+4$$

Determine whether or not the given angles in standard position are coterminal. $$\frac{5 \pi}{12}, \frac{17 \pi}{12}$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\tan t ; \quad g(t)=-\tan t$$

In Exercises \(7-16,\) evaluate all six trigonometric finctions at \(t\) where the given point lies on the terminal side of an angle of \(t\) radians in standard position. $$(1+\sqrt{3}, 1-\sqrt{3})$$

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$2 \pi / 3$$

Determine whether or not the given angles in standard position are coterminal. $$\frac{7 \pi}{6},-\frac{5 \pi}{6}$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\cos t ; \quad g(t)=3 \cos t$$

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$7 \pi / 4$$

Determine whether or not the given angles in standard position are coterminal. $$117^{\circ}, 837^{\circ}$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\sin t ; \quad g(t)=-2 \sin t$$

Exercises \(18-20\) deal with the path of a projectile (such as a baseball, a rocket, or an arrow. If the projectile is fired with an initial velocity of \(v\) feet per second at angle of \(t\) radians and its initial height is k feet, then the path of the projectile is given by $$ y=\left(\frac{-16}{v^{2}} \sec ^{2} t\right) x^{2}+(\tan t) x+k $$ You can think of the projectile as being fired in the direction of the \(x\) -axis from the point ( \(0, k\) ) on the \(y\) -axis. (a) Find a viewing window that shows the path of a projectile that is fired from a 20 -foot high platform at an initial velocity of 120 feet per second at an angle of .8 radians. (b) What is the maximum height reached by the projectile? (c) How far down range does the projectile hit the ground?

Find the rules of the composite functions \(f \circ g\) and \(g \circ f\). $$f(t)=\cos ^{2}(t-2) ; \quad g(t)=5 t+2$$

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$5 \pi / 4$$

Determine whether or not the given angles in standard position are coterminal. $$170^{\circ},-550^{\circ}$$

(a) State the period of the function. (b) Describe the graph of the function between 0 and \(2 \pi\) (c) Find a viewing window that accurately shows exactly four complete waves of the graph. $$f(t)=\sin 200 t$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\sin t ; \quad g(t)=3 \sin t+2$$

Exercises \(18-20\) deal with the path of a projectile (such as a baseball, a rocket, or an arrow. If the projectile is fired with an initial velocity of \(v\) feet per second at angle of \(t\) radians and its initial height is k feet, then the path of the projectile is given by $$ y=\left(\frac{-16}{v^{2}} \sec ^{2} t\right) x^{2}+(\tan t) x+k $$ You can think of the projectile as being fired in the direction of the \(x\) -axis from the point ( \(0, k\) ) on the \(y\) -axis. Do Exercise 18 for a projectile that is fired from ground level at an initial velocity of 80 feet per second at an angle of .4 radians.

Determine if it is possible for a number to satisfy the given conditions. [Hint: Think Pythagorean.] $$\sin t=5 / 13 \text { and } \cos t=12 / 13$$

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$3 \pi / 4$$

Find the radian measure of an angle in standard position that has measure between 0 and \(2 \pi\) and is coterminal with the angle in standard position whose measure is given. $$-\pi / 3$$

List the transformations needed to change the graph of \(f(t)\) into the graph of \(g(t) .\) ISee Section 3.4 .1 $$f(t)=\cos t ; \quad g(t)=5 \cos t+3$$

Exercises \(18-20\) deal with the path of a projectile (such as a baseball, a rocket, or an arrow. If the projectile is fired with an initial velocity of \(v\) feet per second at angle of \(t\) radians and its initial height is k feet, then the path of the projectile is given by $$ y=\left(\frac{-16}{v^{2}} \sec ^{2} t\right) x^{2}+(\tan t) x+k $$ You can think of the projectile as being fired in the direction of the \(x\) -axis from the point ( \(0, k\) ) on the \(y\) -axis. Do Exercise 18 for a projectile that is fired from a 40 -foot high platform at an initial velocity of 125 feet per second at an angle of 1.2 radians.

In Exercises \(15-29,\) find the exact value of the sine, cosine, and tangent of the number, without using a calculator. $$-7 \pi / 3$$

Find the radian measure of an angle in standard position that has measure between 0 and \(2 \pi\) and is coterminal with the angle in standard position whose measure is given. $$-3 \pi / 4$$