91Ó°ÊÓ

(a) The unit circle is the circle centered at _______ with radius _______. (b) The equation of the unit circle is _______. (c) Suppose the point \(P(x, y)\) is on the unit circle. Find the missing coordinate: (i) \(P(1,\text{__})\) (ii) \(P(\text{__}, 1)\) (iii) \(P(-1,\text{__})\) (iv) \(P(\text{__},-1)\)

The sine curve \(y=a \sin k(x-b)\) has amplitude _________ period _______, and horizontal shift. The sine curve \(y=4 \sin 3\left(x-\frac{\pi}{6}\right)\) has amplitude _________. period _______ , and horizontal shift _________.

Find the period, and graph the function. $$y=-3 \sec x$$

An initial amplitude \(k\), damping constant \(c,\) and frequency \(f\) or period \(p\) are given. (Recall that frequency and period are related by the equation \(f=1 / p .\) ) (a) Find a function that models the damped harmonic motion. Use a function of the form \(y=k e^{-c t} \cos \omega t\) in Exercises 21-24 and of the form \(y=k e^{-c t} \sin \omega t\) in Exercises 25-28 (b) Graph the function. $$k=100, \quad c=0.05, \quad p=4$$

Terminal Points Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t\) $$t=\frac{7 \pi}{6}$$

Terminal Points Find the terminal point \(P(x, y)\) on the unit circle determined by the given value of \(t\) $$t=-\frac{7 \pi}{4}$$

The Bay of Fundy in Nova Scotia has the highest tides in the world. In one 12-h period the water starts at mean sea level, rises to 21 ft above, drops to 21 ft below, then returns to mean sea level. Assuming that the motion of the tides is simple harmonic, find an equation that describes the height of the tide in the Bay of Fundy above mean sea level. Sketch a graph that shows the level of the tides over a 12-h period.

Terminal Points and Reference Numbers Find (a) the reference number for each value of \(t\) and (b) the terminal point determined by \(t\) $$t=-\frac{7 \pi}{6}$$

A mass is suspended on a spring. The spring is compressed so that the mass is located 5 cm above its rest position. The mass is released at time \(t=0\) and allowed to oscillate. It is observed that the mass reaches its lowest point \(\frac{1}{2} \mathrm{s}\) after it is released. Find an equation that describes the motion of the mass.

The frequency of oscillation of an object suspended on a spring depends on the stiffness \(k\) of the spring (called the spring constant) and the mass \(m\) of the object. If the spring is compressed a distance \(a\) and then allowed to oscillate, its displacement is given by $$f(t)=a \cos \sqrt{k / m} t$$ (a) A 10 -g mass is suspended from a spring with stiffness \(k=3 .\) If the spring is compressed a distance 5 cm and then released, find the equation that describes the oscillation of the spring. (b) Find a general formula for the frequency (in terms of \(k\) and \(m\) ). (c) How is the frequency affected if the mass is increased? Is the oscillation faster or slower? (d) How is the frequency affected if a stiffer spring is used (larger \(k\) )? Is the oscillation faster or slower?