Chapter 6

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\sin ^{2} t-\tan ^{2} t=-\left(\sin ^{2} t\right)\left(\tan ^{2} t\right)$$

Assume that $$\sin (\pi / 8)=\frac{\sqrt{2-\sqrt{2}}}{2}$$ and use identities to find the exact functional value. $$\tan (\pi / 8)$$

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$\frac{\sec ^{2} t+2 \sec t+1}{\sec t}$$

Convert the given radian measure to degrees. $$-5 \pi / 3$$

The volume \(V(t)\) of air (in cubic inches) in an adult's lungs \(t\) seconds after exhaling is approximately $$V(t)=55+24.5 \sin \left(\frac{\pi x}{2}-\frac{\pi}{2}\right)$$ (a) Find the maximum and minimum amount of air in the lungs. (b) How often does the person exhale? (c) How many breaths per minute does the person take?

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\frac{\sin t}{1+\cos t}=\tan t$$

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$\frac{4 \tan t \sec t+2 \sec t}{6 \sin t \sec t+2 \sec t}$$

Convert the given radian measure to degrees. $$\pi / 45$$

The brightness of the binary star Beta Lyrae (as seen from the earth) varies. Its visual magnitude \(M(t)\) after \(t\) days is approximately $$M(t)=.55 \cos (.97 t)+3.85$$ The visual magnitude scale is reversed from what you would expect: The lower the number, the brighter the star. With this in mind, answer the following questions. (a) Graph the function \(M\) when \(0 \leq t \leq 21\) (b) What is the visual magnitude when the star is brightest? When it is dimmest? (c) What is the period of the magnitude (the interval between its brightest times)?

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\frac{\cos t}{1-\sin t}=\frac{1}{\cos t}+\tan t$$

Assume that $$\sin (\pi / 8)=\frac{\sqrt{2-\sqrt{2}}}{2}$$ and use identities to find the exact functional value. $$\tan (-15 \pi / 8)$$

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$\frac{\sec ^{2} t \csc t}{\csc ^{2} t \sec t}$$

Convert the given radian measure to degrees. $$-\pi / 60$$

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\cos \left(\frac{\pi}{2}+t\right)=-\sin t$$

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$(\sin t+\cos t)(\sin t-\cos t)$$

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$(2+\sqrt{\tan t})(2-\sqrt{\tan t})$$

Convert the given radian measure to degrees. $$-5 \pi / 12$$

In Exercises \(47-54\), find the average rate of change of the function over the given interval. Exact answers are required. $$f(t)=\cos t \text { from } t=\pi / 2 \text { to } t=\pi$$

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\sin \left(\frac{\pi}{2}+t\right)=-\cos t$$

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$(\sin t-\cos t)^{2}$$

In Exercises \(43-48,\) simplify the given expression. Assume that all denominators are nonzero and all quantities under radicals are nonnegative. $$\frac{6 \tan t \sin t-3 \sin t}{9 \sin ^{2} t+3 \sin t}$$

Convert the given radian measure to degrees. $$7 \pi / 15$$

The original Ferris wheel, built by George Ferris for the Columbian Exposition of \(1893,\) was much larger and slower than its modern counterparts: It had a diameter of 250 feet and contained 36 cars, each of which held 60 people; it made one revolution every 10 minutes. Imagine that the Ferris wheel revolves counterclockwise in the \(x-y\) plane with its center at the origin. A car had coordinates (125,0) at time \(t=0 .\) Find the rule of a function that gives the \(y\) -coordinate of the car at time \(t.\)

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\tan t \cos t$$

In Exercises \(49-54\), prove the given identity. $$\tan t=\frac{1}{\cot t}[\text {Hint}: \text { See page } 497]$$

Convert the given radian measure to degrees. $$27 \pi / 5$$

In Exercises \(47-54\), find the average rate of change of the function over the given interval. Exact answers are required. $$g(t)=\sin t \text { from } t=\pi / 6 \text { to } t=11 \pi / 3$$

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\left(\cos ^{2} t-1\right)\left(\tan ^{2} t+1\right)=-\tan ^{2} t$$

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$(\sin t) /(\tan t)$$

In Exercises \(49-54\), prove the given identity. $$\sec (t+2 \pi)=\sec t[\text {Hint}: \text { See page } 500 .]$$

Convert the given radian measure to degrees. $$-41 \pi / 6$$

In Exercises \(47-54\), find the average rate of change of the function over the given interval. Exact answers are required. $$h(t)=\tan t \text { from } t=\pi / 6 \text { to } t=11 \pi / 3$$

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\sqrt{\sin ^{3} t \cos t} \sqrt{\cos t}$$

Determine the positive radian measure of the angle that the second hand of a clock traces out in the given time. 40 seconds

In Exercises \(47-54\), find the average rate of change of the function over the given interval. Exact answers are required. $$f(t)=\cos t \text { from } t=-5 \pi / 4 \text { to } t=\pi / 4$$

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$(\tan t+2)(\tan t-3)-(6-\tan t)+2 \tan t$$

In Exercises \(49-54\), prove the given identity. \(\cot (-t)=-\cot t[\text {Hint}:\) Express the left side in terms of sine and cosine; then use the negative angle identities and express the result in terms of cotangent.]

Determine the positive radian measure of the angle that the second hand of a clock traces out in the given time. 50 seconds

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\left(\frac{4 \cos ^{2} t}{\sin ^{2} t}\right)\left(\frac{\sin t}{4 \cos t}\right)^{2}$$

In Exercises \(49-54\), prove the given identity. $$\sec (-t)=\sec t[\text { Adapt the hint for Exercise } 52 .]$$

In Exercises \(47-54\), find the average rate of change of the function over the given interval. Exact answers are required. $$h(t)=\tan t \text { from } t=\pi / 6 \text { to } t=\pi / 3$$

Initial push is downward from the equilibrium point. [Hint: What does the graph of \(A\) sin \(b t\) look like when \(A<0 ?\)

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\frac{5 \cos t}{\sin ^{2} t} \cdot \frac{\sin ^{2} t-\sin t \cos t}{\sin ^{2} t-\cos ^{2} t}$$

In Exercises \(49-54\), prove the given identity. $$\csc (-t)=-\csc t$$

Determine the positive radian measure of the angle that the second hand of a clock traces out in the given time. 2 minutes and 15 seconds.

In Exercises \(47-54\), find the average rate of change of the function over the given interval. Exact answers are required. $$f(t)=\cos t \text { from } t=\pi / 4 \text { to } t=\pi / 3$$

Graph the function. Does the function appear to be periodic? If so, what is the period? $$f(t)=\cos |t|$$

Weight is pulled 6 centimeters above equilibrium, and the initial movement (at \(t=0\) ) is downward. [Hint: Think cosine. \(]\)

Use algebra and identities in the text to simplify the expression. Assume all denominators are nonzero. $$\frac{\cos ^{2} t+4 \cos t+4}{\cos t+2}$$

In Exercises \(55-60\), find the values of all six trigonometric functions at \(t\) if the given conditions are true. $$\begin{aligned} &\cos t=-1 / 2 \quad \text { and } \quad \sin t>0\\\ &\text { [Hint: }\left.\sin ^{2} t+\cos ^{2} t=1 .\right] \end{aligned}$$