삼각함수 전체 문제

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc \,x$ as $\frac{1}{\sin \,x}.$

$f(x)=\cot (100\pi x)$

For the following exercises, use a graphing calculator to graph two periods of the given function. Note: most graphing calculators do not have a cosecant button; therefore, you will need to input $\csc \,x$ as $\frac{1}{\sin \,x}.$

$f(x)={\sin }^{2}x+{\cos }^{2}x$

The function $f(x)=20\tan \left(\frac{\pi }{10}x\right)$ marks the distance in the movement of a light beam from a police car across a wall for time $x,$ in seconds, and distance $f(x),$ in feet.

• ⓐ Graph on the interval $[0,\,5].$
• ⓑ Find and interpret the stretching factor, period, and asymptote.
• ⓒ Evaluate $f(1)$ and $f(2.5)$ and discuss the function’s values at those inputs.

Standing on the shore of a lake, a fisherman sights a boat far in the distance to his left. Let $x,$ measured in radians, be the angle formed by the line of sight to the ship and a line due north from his position. Assume due north is 0 and $x$ is measured negative to the left and positive to the right. (See (the referenced figure).) The boat travels from due west to due east and, ignoring the curvature of the Earth, the distance $d(x),$ in kilometers, from the fisherman to the boat is given by the function $d(x)=1.5\sec (x).$

• ⓐWhat is a reasonable domain for $d(x)?$
• ⓑGraph $d(x)$ on this domain.
• ⓒFind and discuss the meaning of any vertical asymptotes on the graph of $d(x).$
• ⓓCalculate and interpret $d\left(-\frac{\pi }{3}\right).$ Round to the second decimal place.
• ⓔCalculate and interpret $d\left(\frac{\pi }{6}\right).$ Round to the second decimal place.
• ⓕWhat is the minimum distance between the fisherman and the boat? When does this occur?

A laser rangefinder is locked on a comet approaching Earth. The distance $g(x),$ in kilometers, of the comet after $x$ days, for $x$ in the interval 0 to 30 days, is given by $g(x)=250,000\csc \left(\frac{\pi }{30}x\right).$

• ⓐ Graph $g(x)$ on the interval $[0,\,30].$
• ⓑ Evaluate $g(5)$ and interpret the information.
• ⓒ What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond?
• ⓓ Find and discuss the meaning of any vertical asymptotes.

A video camera is focused on a rocket on a launching pad 2 miles from the camera. The angle of elevation from the ground to the rocket after $x$ seconds is $\frac{\pi }{120}x.$

• ⓐWrite a function expressing the altitude $h(x),$ in miles, of the rocket above the ground after $x$ seconds. Ignore the curvature of the Earth.
• ⓑGraph $h(x)$ on the interval $(0,\,60).$
• ⓒEvaluate and interpret the values $h(0)$ and $h(30).$
• ⓓWhat happens to the values of $h(x)$ as $x$ approaches 60 seconds? Interpret the meaning of this in terms of the problem.

For the following exercises, find the angle $\theta$ in the given right triangle. Round answers to the nearest hundredth.

For the following exercises, find the angle $\theta$ in the given right triangle. Round answers to the nearest hundredth.

Graph $y={\sin }^{-1}x$ and state the domain and range of the function.

Graph $y=\arccos \,x$ and state the domain and range of the function.

Graph one cycle of $y={\tan }^{-1}x$ and state the domain and range of the function.

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

$f(x)=-3\cos \,x+3$

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

$f(x)=\frac{1}{4}\sin \,x$

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

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

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

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

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

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

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

$f(x)=2\left(\cos \left(x-\frac{4\pi }{3}\right)+1\right)$

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

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

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

$f(x)=-100\sin (50x-20)$

For the following exercises, graph the functions for two periods and determine the amplitude or stretching factor, period, midline equation, and asymptotes.

$f(x)=\tan \,x-4$