Chapter 6

Convert the given degree measure to radians. $$135^{\circ}$$

In Exercises \(31-36\), write the expression as a single real mum. ber. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\cos (\pi / 2) \cos (\pi / 4)-\sin (\pi / 2) \sin (\pi / 4)$$

Graph the function over the interval \([0,2 \pi)\) and determine the location of all local maxima and minima. [This can be done either graphically or algebraically.] $$g(t)=2 \sin (2 t / 3-\pi / 9)$$

Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) $$\cos t=2 / 3$$

In Exercises \(30-36,\) perform the indicated operations, then simplify your answers by using appropriate definitions and identities. $$(\sin t-\csc t)^{2}$$

Convert the given degree measure to radians. $$-165^{\circ}$$

In Exercises \(31-36\), write the expression as a single real mum. ber. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\cos (2 \pi / 3) \cos (\pi)+\sin (2 \pi / 3) \sin (\pi)$$

Graph the function over the interval \([0,2 \pi)\) and determine the location of all local maxima and minima. [This can be done either graphically or algebraically.] $$f(t)=-2 \sin (3 t-\pi)$$

Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) $$\sin t=-1 / 2$$

In Exercises \(30-36,\) perform the indicated operations, then simplify your answers by using appropriate definitions and identities. $$(\cot t-\tan t)\left(\cot ^{2} t+1+\tan ^{2} t\right)$$

Convert the given degree measure to radians. $$-225^{\circ}$$

In Exercises \(31-36\), write the expression as a single real mum. ber. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\sin (3 \pi / 4) \cos (5 \pi / 6)-\cos (3 \pi / 4) \sin (5 \pi / 6)$$

Graph the function over the interval \([0,2 \pi)\) and determine the location of all local maxima and minima. [This can be done either graphically or algebraically.] $$h(t)=\frac{1}{2} \cos \left(\frac{\pi}{2} t-\frac{\pi}{8}\right)+1$$

Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) \(\sin t=k,\) where \(k\) is a nonzero constant such that \(-1

(a) Show that \(\tan (t+2 \pi)=\tan t\) for every \(t\) in the domain of tan \(t .\) [ Hint: Use the definition of tangent and some identities proved in the text. \(]\) (b) Verify that it appears true that \(\tan (x+\pi)=\tan x\) for every \(t\) in the domain by using your calculator's table feature to make a table of values for \(y_{1}=\tan (x+\pi)\) and \(y_{2}=\tan x\)

In Exercises \(30-36,\) perform the indicated operations, then simplify your answers by using appropriate definitions and identities. $$(\sin t+\csc t)\left(\sin ^{2} t+\csc ^{2} t-1\right)$$

Convert the given degree measure to radians. $$252^{\circ}$$

In Exercises \(31-36\), write the expression as a single real number. Do not use decimal approximations. IHint: Exercises \(15-21 \text { may be helpful. }]\) $$\sin (-7 \pi / 3) \cos (5 \pi / 4)+\cos (-7 \pi / 3) \sin (5 \pi / 4)$$

Graph \(f(t)\) in a viewing window with \(-2 \pi \leq t \leq 2 \pi .\) Use a maximum finder and a root finder to determine constants \(A, b, c\) such that the graph of \(f(t)\) appears to coincide with the graph of \(g(t)=A \sin (b t+c).\) $$f(t)=3 \sin t+2 \cos t$$

Use the graphs of the trigonometric functions to determine the number of solutions of the equation between 0 and \(2 \pi\) \(\cos t=k,\) where \(k\) is a constant such that \(-1

Assume that $$\cos t=-2 / 5 \quad \text { and } \quad \pi

In Exercises \(37-42\), factor and simplify the given expression. $$\sec t \csc t-\csc ^{2} t$$

Convert the given degree measure to radians. $$930^{\circ}$$

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(3,5)$$

Assume that $$\cos t=-2 / 5 \quad \text { and } \quad \pi

In Exercises \(37-42\), factor and simplify the given expression. $$\tan ^{2} t-\cot ^{2} t$$

Convert the given degree measure to radians. $$-585^{\circ}$$

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(-2,1)$$

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\sin (-t)=-\sin t$$

Assume that $$\cos t=-2 / 5 \quad \text { and } \quad \pi

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

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(-4,-5)$$

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\cos (-t)=\cos t$$

Graph \(f(t)\) in a viewing window with \(-2 \pi \leq t \leq 2 \pi .\) Use a maximum finder and a root finder to determine constants \(A, b, c\) such that the graph of \(f(t)\) appears to coincide with the graph of \(g(t)=A \sin (b t+c).\) $$f(t)=2 \sin (3 t-5)-3 \cos (3 t+2)$$

Assume that $$\cos t=-2 / 5 \quad \text { and } \quad \pi

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

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(3,-4)$$

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

Convert the given radian measure to degrees. $$-\pi / 10$$

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(\sqrt{3},-8)$$

Use graphs to determine whether the equation could possibly be an identity or definitely is not an identity. $$\sin (t+\pi)=-\sin t$$

Assume that $$\cos t=-2 / 5 \quad \text { and } \quad \pi

In Exercises \(37-42\), factor and simplify the given expression. $$\csc ^{4} t+4 \csc ^{2} t-5$$

Convert the given radian measure to degrees. $$2 \pi / 5$$

In Exercises \(37-42,\) find \(\sin t,\) cos \(t,\) tan \(t\) when the terminal side of an angle of t radians in standard position passes through the given point. $$(-2, \pi)$$

Do parts (a) and (b) of Example 9 for a person whose blood pressure is given by $$g(t)=21 \cos (2.5 \pi t)+113$$ According to current guidelines, someone with systolic pressure above 140 or diastolic pressure above 90 has high blood pressure and should see a doctor about it. What would you advise the person in this case?

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

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

Convert the given radian measure to degrees. $$3 \pi / 4$$

$$\text { In Exercises } 43-46, \text { use a calculator in radian mode.}$$ When a plane flies faster than the speed of sound, the sound waves it generates trail the plane in a cone shape, as shown in the figure. When the bottom part of the cone hits the ground, you hear a sonic boom. The equation that describes this situation is $$ \sin \left(\frac{t}{2}\right)=\frac{w}{p} $$ where \(t\) is the radian measure of the angle of the cone, \(w\) is the speed of the sound wave, \(p\) is the speed of the plane, and \(p>w\) (a) Find the speed of the sound wave when the plane flies at 1200 mph and \(t=.8\) (b) Find the speed of the plane if the sound wave travels at \(500 \mathrm{mph}\) and \(t=.7\) (IMAGE CAN'T COPY)