6.3 Inverse trigonometric functions  (Page 9/15)

What are the amplitude, period, and phase shift for the function?

Over this domain, when does the population reach 18,000? 13,000?

What is the predicted population in 2007? 2010?

Suppose the graph of the displacement function is shown in [link] , where the values on the x -axis represent the time in seconds and the y -axis represents the displacement in inches. Give the equation that models the vertical displacement of the weight on the spring.

At time = 0, what is the displacement of the weight?

At what time does the displacement from the equilibrium point equal zero?

What is the time required for the weight to return to its initial height of 5 inches? In other words, what is the period for the displacement function?

${\sin }^{-1}\left(1\right)$

${\cos }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

${\tan }^{\mathrm{-1}}\left(\mathrm{-1}\right)$

${\cos }^{-1}\left(\frac{1}{\sqrt{2}}\right)$

${\sin }^{-1}\left(\frac{-\sqrt{3}}{2}\right)$

${\sin }^{-1}\left(\cos \left(\frac{\pi }{6}\right)\right)$

${\cos }^{-1}\left(\tan \left(\frac{3\pi }{4}\right)\right)$

$\sin \left({\sec }^{-1}\left(\frac{3}{5}\right)\right)$

$\cot \left({\sin }^{-1}\left(\frac{3}{5}\right)\right)$

$\tan \left({\cos }^{-1}\left(\frac{5}{13}\right)\right)$

$\sin \left({\cos }^{-1}\left(\frac{x}{x+1}\right)\right)$

Graph $f\left(x\right)=\cos x$ and $f\left(x\right)=\sec x$ on the interval $\left[0,2\pi \right)$ and explain any observations.

Graph $f(x)=\sin x$ and $f\left(x\right)=\csc x$ and explain any observations.

Graph the function $f\left(x\right)=\frac{x}{1}-\frac{x^3}{3!}+\frac{x^5}{5!}-\frac{x^7}{7!}$ on the interval $\left[-1,1\right]$ and compare the graph to the graph of $f\left(x\right)=\sin x$ on the same interval. Describe any observations.

$f\left(x\right)=0.5\sin x$

$f\left(x\right)=5\cos x$

$f\left(x\right)=5\sin x$

$f\left(x\right)=\sin \left(3x\right)$

$f\left(x\right)=-\cos \left(x+\frac{\pi }{3}\right)+1$

$f\left(x\right)=5\sin \left(3\left(x-\frac{\pi }{6}\right)\right)+4$

$f\left(x\right)=3\cos \left(\frac{1}{3}x-\frac{5\pi }{6}\right)$

$f\left(x\right)=\tan \left(4x\right)$

$f\left(x\right)=-2\tan \left(x-\frac{7\pi }{6}\right)+2$

$f\left(x\right)=\pi \cos \left(3x+\pi \right)$

$f\left(x\right)=5\csc \left(3x\right)$

$f\left(x\right)=\pi \sec \left(\frac{\pi }{2}x\right)$

$f\left(x\right)=2\csc \left(x+\frac{\pi }{4}\right)-3$

Give in terms of a sine function.

Give in terms of a sine function.

Give in terms of a tangent function.

$y=\sin \left(\frac{\pi }{6}x+\pi \right)-3$

$y=8\sin \left(\frac{7\pi }{6}x+\frac{7\pi }{2}\right)+6$

The outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the temperature is 68°F at midnight and the high and low temperatures during the day are 80°F and 56°F, respectively. Assuming $t$ is the number of hours since midnight, find a function for the temperature, $D,$ in terms of $t.$

Water is pumped into a storage bin and empties according to a periodic rate. The depth of the water is 3 feet at its lowest at 2:00 a.m. and 71 feet at its highest, which occurs every 5 hours. Write a cosine function that models the depth of the water as a function of time, and then graph the function for one period.

$g\left(x\right)=3\tan \left(6x+42\right)$

$n\left(x\right)=4\csc \left(\frac{5\pi }{3}x-\frac{20\pi }{3}\right)$

Write the equation for the graph in [link] in terms of the secant function and give the period and phase shift.

If $\tan x=3,$ find $\tan \left(-x\right).$

If $\sec x=4,$ find $\sec \left(-x\right).$

Graph $m\left(x\right)=\sin \left(2x\right)+\cos \left(3x\right)$ on the viewing window $\left[-10,10\right]$ by $\left[-3,3\right].$ Approximate the graph’s period.

Graph $n\left(x\right)=0.02\sin \left(50\pi x\right)$ on the following domains in $x:$ $\left[0,1\right]$ and $\left[0,3\right].$ Suppose this function models sound waves. Why would these views look so different?

Graph $f\left(x\right)=\frac{\sin x}{x}$ on $\left[-0.5,0.5\right]$ and explain any observations.

What is the largest possible value for $f\left(x\right)?$

What is the smallest possible value for $f\left(x\right)?$

Where is the function increasing on the interval $\left[0,2\pi \right]?$

Sine curve with amplitude 3, period $\frac{\pi }{3},$ and phase shift $\left(h,k\right)=\left(\frac{\pi }{4},2\right)$

Cosine curve with amplitude 2, period $\frac{\pi }{6},$ and phase shift $\left(h,k\right)=\left(-\frac{\pi }{4},3\right)$

$f\left(x\right)=5\cos \left(3x\right)+4\sin \left(2x\right)$

$f\left(x\right)=e^{\sin t}$

${\sin }^{-1}\left(\frac{\sqrt{3}}{2}\right)$

${\tan }^{-1}\left(\sqrt{3}\right)$

${\cos }^{-1}\left(-\frac{\sqrt{3}}{2}\right)$

${\cos }^{-1}\left(\sin \left(\pi \right)\right)$

${\cos }^{-1}\left(\tan \left(\frac{7\pi }{4}\right)\right)$

$\cos \left({\sin }^{-1}\left(1-2x\right)\right)$

${\cos }^{-1}\left(-0.4\right)$

$\cos \left({\tan }^{-1}\left(x^2\right)\right)$

$\tan t$

$\csc t$

Given [link] , find the measure of angle $\theta$ to three decimal places. Answer in radians.

$\arcsin \left(\sin \left(\frac{5\pi }{6}\right)\right)=\frac{5\pi }{6}$

$\arccos \left(\cos \left(\frac{5\pi }{6}\right)\right)=\frac{5\pi }{6}$

The grade of a road is 7%. This means that for every horizontal distance of 100 feet on the road, the vertical rise is 7 feet. Find the angle the road makes with the horizontal in radians.