삼각함수

For the following exercises, find a. the amplitude, b. the period, and c. the phase shift with direction for each function.

$y=4\cos \left(2x-\frac{\pi }{2}\right)$

[T] Find the length of the arc intercepted by central angle $\theta$ in a circle of radius r. Round to the nearest hundredth.

a. $r=12.8$ cm, $\theta =\frac{5\pi }{6}$ rad b. $r=4.378$ cm, $\theta =\frac{7\pi }{6}$ rad c. $r=0.964$ cm, $\theta =50^\circ$ d. $r=8.55$ cm, $\theta =325^\circ$

[T] As a point P moves around a circle, the measure of the angle changes. The measure of how fast the angle is changing is called angular speed, $\omega ,$ and is given by $\omega =\theta /t,$ where $\theta$ is in radians and t is time. Find the angular speed for the given data. Round to the nearest thousandth.

a. $\theta =\frac{7\pi }{4}\text{rad},t=10$ sec b. $\theta =\frac{3\pi }{5}\text{rad},t=8$ sec c. $\theta =\frac{2\pi }{9}\text{rad},t=1$ min d. $\theta =23.76\text{rad,}t=14$ min

[T] A total of 250,000 m² of land is needed to build a nuclear power plant. Suppose it is decided that the area on which the power plant is to be built should be circular.

a. Find the radius of the circular land area. b. If the land area is to form a $45^\circ$ sector of a circle instead of a whole circle, find the length of the curved side.

[T] The area of an isosceles triangle with equal sides of length x is

$\frac{1}{2}{x}^{2}\sin \theta ,$

where $\theta$ is the angle formed by the two sides. Find the area of an isosceles triangle with equal sides of length 8 in. and angle $\theta =5\pi /12$ rad.

[T] A particle travels in a circular path at a constant angular speed $\omega .$ The angular speed is modeled by the function $\omega =9|\cos (\pi t-\pi /12)|.$ Determine the angular speed at $t=9$ sec.

[T] An alternating current for outlets in a home has voltage given by the function

$V(t)=150\cos 368t,$

where V is the voltage in volts at time t in seconds.

a. Find the period of the function and interpret its meaning. b. Determine the number of periods that occur when 1 sec has passed.

[T] The number of hours of daylight in a northeast city is modeled by the function

$$N(t)=12+3\sin \left[\frac{2\pi }{365}(t-79)\right],$$

where t is the number of days after January 1.

a. Find the amplitude and period. b. Determine the number of hours of daylight on the longest day of the year. c. Determine the number of hours of daylight on the shortest day of the year. d. Determine the number of hours of daylight 90 days after January 1. e. Sketch the graph of the function for one period starting on January 1.

[T] Suppose that $T=50+10\sin \left[\frac{\pi }{12}(t-8)\right]$ is a mathematical model of the temperature (in degrees Fahrenheit) at t hours after midnight on a certain day of the week.

a. Determine the amplitude and period. b. Find the temperature 7 hours after midnight. c. At what time does $T=60^\circ ?$ d. Sketch the graph of $T$ over $0\le t\le 24.$

[T] The function $H(t)=8\sin \left(\frac{\pi }{6}t\right)$ models the height H (in feet) of the tide t hours after midnight. Assume that $t=0$ is midnight.

a. Find the amplitude and period. b. Graph the function over one period. c. What is the height of the tide at 4:30 a.m.?