Chapter 6

Why are the sine and cosine functions called periodic functions?

How does the graph of \(y=\sin x\) compare with the graph of \(y=\cos x ?\) Explain how you could horizontally translate the graph of \(y=\sin x\) to obtain \(y=\cos x .\)

For the equation \(A \cos (B x+C)+D,\) what constants affect the range of the function and how affect the range?

How does the range of a translated sine function relate to the equation \(y=A \sin (B x+C)+D ?\)

How can the unit circle be used to construct the graph of \(f(t)=\sin t ?\)

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=2 \sin x $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=\frac{2}{3} \cos x $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=-3 \sin x $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=4 \sin x $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=2 \cos x $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=\cos (2 x) $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=2 \sin \left(\frac{1}{2} x\right) $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=4 \cos (\pi x) $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ f(x)=3 \cos \left(\frac{6}{5} x\right) $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ y=3 \sin (8(x+4))+5 $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ y=2 \sin (3 x-21)+4 $$

For the following exercises, graph two full periods of each function and state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0 .\) Round answers to two decimal places if necessary. $$ y=5 \sin (5 x+20)-2 $$

For the following exercises, graph one full period of each function, starting at \(x=0 .\) For each function, state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0\) . State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary. $$f(t)=2 \sin \left(t-\frac{5 \pi}{6}\right)$$

For the following exercises, graph one full period of each function, starting at \(x=0 .\) For each function, state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0\) . State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary. $$ f(t)=-\cos \left(t+\frac{\pi}{3}\right)+1 $$

For the following exercises, graph one full period of each function, starting at \(x=0 .\) For each function, state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0\) . State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary. $$ f(t)=4 \cos \left(2\left(t+\frac{\pi}{4}\right)\right)-3 $$

For the following exercises, graph one full period of each function, starting at \(x=0 .\) For each function, state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0\) . State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary. $$ f(t)=-\sin \left(\frac{1}{2} t+\frac{5 \pi}{3}\right) $$

For the following exercises, graph one full period of each function, starting at \(x=0 .\) For each function, state the amplitude, period, and midine. State the maximum and minimum \(y\) -values and their corresponding \(x\) -values on one period for \(x>0\) . State the phase shift and vertical translation, if applicable. Round answers to two decimal places if necessary. $$ f(x)=4 \sin \left(\frac{\pi}{2}(x-3)\right)+7 $$

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi),\) solve \(f(x)=0\)

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi),\) solve \(f(x)=\frac{1}{2}\)

For the following exercises, let \(f(x)=\sin x\) Evaluate \(f\left(\frac{\pi}{2}\right)\)

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi), f(x)=\frac{\sqrt{2}}{2} .\) Find all values of \(x\)

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi),\) the maximum value(s) of the function occur(s) at what \(x\) -value(s)?

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi),\) the minimum value(s) of the function occur(s) at what \(x\) -value(s)?

For the following exercises, let \(f(x)=\sin x\) Show that \(f(-x)=-f(x) .\) This means that \(f(x)=\sin x\) is an odd function and possesses symmetry with respect to _________.

For the following exercises, let \(f(x)=\cos x\) On \([0,2 \pi),\) solve the equation \(f(x)=\cos x=0\)

For the following exercises, let \(f(x)=\cos x\) On \([0,2 \pi),\) solve \(f(x)=\frac{1}{2}\)

For the following exercises, let \(f(x)=\cos x\) On \([0,2 \pi),\) find the \(x\) -intercepts of \(f(x)=\cos x\)

For the following exercises, let \(f(x)=\cos x\) On \([0,2 \pi),\) find the \(x\) -values at which the function has a maximum or minimum value.

For the following exercises, let \(f(x)=\cos x\) On \([0,2 \pi),\) solve the equation \(f(x)=\frac{\sqrt{3}}{2}\)

For the following exercises, let \(f(x)=\cos x\) Graph \(h(x)=x+\sin x\) on \([0,2 \pi] .\) Explain why the graph appears as it does.

For the following exercises, let \(f(x)=\cos x\) Graph \(f(x)=\frac{\sin x}{x}\) on the window \([-5 \pi, 5 \pi]\) and explain what the graph shows.

A Ferris wheel is 25 meter in diameter and boarded from a platform that is 1 meter above the ground. The six o'clock position on the Ferris wheel is level with the loading plafform. The wheel completes 1 full revolution in 10 minutes. The function \(h(t)\) gives a person's height in meters above the ground \(t\) minutes after the wheel begins to tum. a. Find the amplitude, midline, and period of \(h(t) .\) b. Find a formula for the height function \(h(t)\) c. How high off the ground is a person after 5 minutes?

Explain how the graph of the sine function can be used to graph \(y=\csc x\)

Explain why the period of \(\tan x\) is equal to \(\pi\)

Why are there no intercepts on the graph of \(y=\csc x ?\)

How does the period of \(y=\csc x\) compare with the period of \(y=\sin x ?\)

For the following exercises, find the period and horizontal shift of each of the functions. $$ f(x)=2 \tan (4 x-32) $$

For the following exercises, find the period and horizontal shift of each of the functions. $$ h(x)=2 \sec \left(\frac{\pi}{4}(x+1)\right) $$

For the following exercises, find the period and horizontal shift of each of the functions. $$ m(x)=6 \csc \left(\frac{\pi}{3} x+\pi\right) $$

For the following exercises, find the period and horizontal shift of each of the functions. If \(\tan x=-1.5,\) find \(\tan (-x)\)

For the following exercises, find the period and horizontal shift of each of the functions. If \(\sec x=2,\) find \(\sec (-x)\)

For the following exercises, find the period and horizontal shift of each of the functions. If \(\csc x=-5,\) find \(\csc (-x)\)

For the following exercises, rewrite each expression such that the argument \(x\) is positive. $$\cot (-x) \cos (-x)+\sin (-x)$$

For the following exercises, rewrite each expression such that the argument \(x\) is positive. $$\cos (-x)+\tan (-x) \sin (-x)$$

For the following exercises, sketch two periods of the graph for each of the following functions. Identify the stretching factor, period, and asymptotes. $$ f(x)=2 \tan (4 x-32) $$