Chapter 19

(a) Convert the following to radians. (i) \(60^{\circ}\) (ii) \(30^{\circ}\) (iii) \(45^{\circ}\) (iv) \(-120^{\circ}\) (b) Convert 2 radians to degrees.

(a) On the same set of axes graph the following. Set the domain to show at least one complete cycle of the function. (Colored pens/pencils can be helpful in identifying which graph is which.) (i) \(y=\sin x\) (ii) \(y=2 \sin x\) (iii) \(y=-3 \sin x\) (b) Describe in words the effect of the parameter \(A\) in \(y=A \sin (x)\).

(a) Using what you know about the properties of polynomial functions, explain how the graph of \(f(x)=\sin x\) tells you that it is not a polynomial. (Think about the number of roots and the long-term behavior.) (b) Using what you know about the properties of rational functions, explain how the graph of \(f(x)=\tan x\) tells you that it is not a rational function. (Think about the number of roots and vertical asymptotes.) (c) What are characteristics of trigonometric functions that distinguish them from other functions we've studied?

Convert these angles to radian measure. (a) \(-60^{\circ}\) (b) \(45^{\circ}\) (c) \(-270^{\circ}\) (d) \(40^{\circ}\) (e) \(-120^{\circ}\)

Use the calibrated unit circle to estimate all \(t\) -values between 0 and 6 such that (a) \(\cos t=0.3\). (b) \(\sin t=0.7\). (c) \(\sin t=-0.7\).

(a) On the same set of axes graph the following. (i) \(y=\sin x\) (ii) \(y=\sin (2 x)\) (iii) \(y=\sin (x / 2)\) (iv) \(y=\sin (4 x)\) (v) \(y=\sin (-2 x)\) (b) Describe in words the effect of the parameter \(B\) in \(y=\sin (B x)\).

Evaluate the following limits. (a) \(\lim _{x \rightarrow-\pi / 2^{+}} \tan x\) (b) \(\lim _{x \rightarrow-\pi / 2^{-}} \tan x\) (c) \(\lim _{x \rightarrow-\pi / 2} \tan x\)

Convert these angles given in radians to degrees. (a) \(\frac{3 \pi}{4}\) (b) \(\frac{-3 \pi}{4}\) (c) \(\frac{5 \pi}{6}\) (d) \(\frac{3 \pi}{2}\) (e) \(\frac{5 \pi}{4}\) (f) \(-3.2\) (g) 4

(a) Describe in words the effect of the parameter \(C\) in \(y=\sin (x)+C\). (b) Describe in words the effect of the parameter \(D\) in \(y=\sin (x+D)\).

A second hand of a clock is 6 inches long. (a) How far does the pointer of the second hand travel in 20 seconds? (b) How far does the pointer of the second hand travel when the second hand travels through an angle of \(70^{\circ}\) ? (c) In one hour the minute hand of the clock moves through an angle of \(2 \pi\) radians. In this amount of time, through what angle does the second hand travel? The hour hand? Give your answers in radians.

Beginning at point \((1,0)\) and traveling a distance \(t\) counterclockwise along the unit circle, we arrive at a point with coordinates \(\left(\frac{-1}{3}, \frac{2 \sqrt{2}}{3}\right)\). Find the following. (a) \(\cos t\) (b) \(\sin t\) (c) \(\sin (-t)\) (d) \(\cos (-t)\) (e) \(\sin (t-\pi)\) (f) \(\sin (t-10 \pi)\) (g) Is \(\sin \left(t+\frac{\pi}{2}\right)\) positive, negative, or zero? Explain.

Suppose \(\tan \alpha=b\). Find the following. Explain your reasoning. (a) \(\tan (\alpha+\pi)\) (b) \(\tan (-\alpha)\) (c) \(\tan (\pi-\alpha)\)

A bicycle gear with radius 4 inches is rotating with a frequency of 50 revolutions per minute. In 2 minutes what distance has been covered by a point on the corresponding chain?

Which of the following equations hold for all \(x ?\) Explain your answers in terms of the unit circle. (a) \(\sin x=\sin (-x)\) (b) \(\sin x=-\sin (-x)\) (c) \(\cos x=\cos (-x)\) (d) \(\cos x=-\cos (-x)\)

Find the domain and range of each of the following functions. (a) \(f(x)=3 \sin (2 x+1)\) (b) \(g(x)=|2 \cos x|\) (c) \(h(x)=\cos |x|\) (d) \(j(x)=2 \cos x-1\) (c) \(k(x)=\sqrt{\sin x}\)

Graph the following. (a) \(g(x)=-\tan x\) (b) \(h(x)=|\tan x|\)

A nautical mile is the distance along the surface of the earth subtended by an angle with vertex at the center of the earth and measuring \(\frac{1}{60}^{\circ}\). (a) The radius of the earth is about 3960 miles. Use this to approximate a nautical mile. Give your answer in feet. (One mile is 5280 feet.) (b) The Random House Dictionary defines a nautical mile to be 6076 feet. Use this to get a more accurate estimate for the radius of the earth than that given in part (a).

Evaluate the following limits. Explain your reasoning. (a) \(\lim _{x \rightarrow \infty} \sin x\) (b) \(\lim _{x \rightarrow \infty} \frac{\sin x}{x}\)

Find all \(x\) such that \(\tan x=0\).

The earth travels around the sun in an elliptical orbit, but the ellipse is very close to circular. The earth's distance from the sun varies between 147 million kilometers at perihelion (when the earth is closest to the sun) and 153 million kilometers at aphelion (when the earth is farthest from the sun). Use the following simplifying assumptions to give a rough estimate of how far the earth travels along its orbit each day. Simplifying assumptions: Model the earth's path around the sun as a circle with radius 150 million \(\mathrm{km}\). Assume that the earth completes a trip around the circle every 365 days.

Evaluate the following limits. (a) \(\lim _{x \rightarrow \infty} \cos x\) (b) \(\lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)\) (c) \(\lim _{x \rightarrow 0^{+}} \frac{1}{\sin x}\) (d) \(\lim _{x \rightarrow \infty} \sin \left(\frac{x^{2}}{x+1}\right)\) (e) \(\lim _{x \rightarrow \infty} \cos \left(\frac{\pi x^{3}-99}{x^{3}-x^{2}+7}\right)\)

Find all \(x\) such that (a) \(\tan x=1\). (b) \(\tan x=-1\). Try to do this using the unit circle definitions.

A bicycle wheel is 26 inches in diameter. When the brakes are applied the bike wheel makes \(2.2\) revolutions before coming to a halt. How far has the bike traveled? (Assume the bike does not skid.)

Graph the following. (a) \(y=|\sin x|\) (b) \(y=\sin |x|\)

Sketch the graph of \(g(x)=\tan 2 x\) on \([0,2 \pi]\).

Find the period, amplitude, and balance value of each of the following functions. (a) \(f(x)=0.5 \sin (3 x)\) (b) \(g(x)=-4 \cos (x / 3)\) (c) \(h(x)=\frac{\sin (0.2 x)+\pi}{\pi}\) (d) \(j(x)=4[\sin (\pi x)-1]\) (c) \(k(x)=4 \sin (\pi x-1)\)

Sketch the graph of \(f(x)=3 \tan \left(\frac{x}{2}\right)\) on the interval \([-2 \pi, 2 \pi]\).

(a) A Ferris wheel with diameter 20 feet makes one revolution every 8 minutes. Graph the height of a point on the Ferris wheel versus time, assuming that at \(t=0\) the point is at height \(0 .\) Give an equation whose graph is the picture you've drawn. (b) The Ferris wheel slows down so that it makes one revolution every 10 minutes. Adjust both your picture and your equation.

Consider the function \(f(x)=\frac{\cos x}{\sin x}\). (a) Where is \(f\) undefined? (b) Where are the zeros of \(f ?\) (c) What is the period of \(f ?\) (d) Sketch the graph of \(f\) on the interval \([0,2 \pi]\).

If \(\sin \theta=\frac{5}{13}\) and \(\cos \theta\) is negative, label the coordinates of the points \(P(\theta), P(-\theta)\) and \(P(\pi-\theta)\) on the unit circle. Then find the following. (a) \(\cos \theta\) (b) \(\tan \theta\) (c) \(\cos (-\theta)\) (d) \(\sin (\theta+\pi)\) (e) \(\tan (\pi-\theta)\)

(a) What is the domain of \(g(x)=\sqrt{\sin x}\) ? (b) On the same set of axes graph \(f(x)=\sin x\) and \(g(x)=\sqrt{\sin x}\). (Use different colors if you have them at your disposal.)

Sketch the graph of \(f(x)=\frac{1}{\sin x}\) on \([0,2 \pi]\). What is the period of \(f ?\)

Suppose we have the equation of a sine curve, \(y=\sin \left(\frac{\pi x}{2}\right)\), with period 4, amplitude 1 . (a) We wish to shift the graph over 1 unit to the left. Which of the following will accomplish this? i. \(y=\sin \left(\frac{\pi}{2} x-1\right)\) ii. \(y=\sin \left(\frac{\pi}{2}(x-1)\right)=\sin \left(\frac{\pi}{2} x-\frac{\pi}{2}\right)\) iii. \(y=\sin \left(\frac{\pi}{2} x+1\right)\) iv. \(y=\sin \left(\frac{\pi}{2}(x+1)\right)=\sin \left(\frac{\pi}{2} x+\frac{\pi}{2}\right)\) Be sure that you get this right. Draw a picture of what you are aiming for, and then try a point or two to verify. For example, you might run a test on \(x=0\). (b) For each of the remaining three choices in part (a), describe in words what happens to the original graph. (c) In order to obtain the desired result in part (a), it is probably simplest to use a cosine function. What cosine function will give the desired result? (d) Suppose \(A, B\), and \(C\) are positive constants and \(y=A \sin (B x+C) .\) What are the period and amplitude of the sine graph? Describe the horizontal shift.

Suppose \(\tan \beta=7\) (a) Find all \(x\) such that \(\tan x=7\). (b) Find all \(x\) such that \(\tan x=-7\).

Decide whether each of the following functions is even, odd, or neither. (a) \(f(x)=1+\cos x\) (b) \(g(x)=1+\sin x\) (c) \(h(x)=\sin 2 x+\tan x\) (d) \(j(x)=|\sin x|\) (e) \(k(x)=\sin x+\cos x\)

The region of Bogor in Java has a rainforest climate with \(450 \mathrm{~mm}\) of rain falling in the rainiest month (February) and \(230 \mathrm{~mm}\) of rain falling in the driest month. Model the amount of rain per month using a sinusoidal function. Let \(R(t)\) be the number of millimeters of rain per month where \(t=0\) denotes the height of the rainy season and is measured in months. (a) Give an expression for \(R(t)\). (b) According to your model, on average how many millimeters of rain fall per month each year? Recall that the balance value of the function is the average value. (c) How many millimeters of rain does your model predict each year? Compare this with the figure of \(4370 \mathrm{~mm}\) recorded. What is the percent error involved in your model?

Studies conducted over a nine-year period indicate that in the alpine belt of the tropics of Venezuela, in Páramo de Mucuchies, the number of rainy days per month varies from an average low of 4 per month in the dry season to a high of 23 per month in the wet season, half a year later. (a) Model the number of rainy days per month using a sinusoidal function. Let \(t=0\) correspond to the driest month. (b) On average, how many rainy days does your model predict per year? Compare this with the recorded average number of rainy days per year: \(181 .\) (Your estimate will be a bit low, because in fact the rainy season is slightly longer than the dry season.)

A wave has amplitude 3 and frequency \(10 .\) Give a possible formula for the wave. (There are infinitely many correct answers.)

The gravitational pull of the sun and moon on large bodies of water produces tides. Tides generally rise and fall twice every 25 hours. (The length of a cycle is \(12.5\) hours as opposed to 12 hours due to the moon's revolution around the earth.) The range between high and low tide varies greatly with location. On the Pacific coast of America this range can be as much as 15 feet. The Bay of Fundy in New Brunswick has an extremely dramatic range of about 45 feet. (a) Model the tidal fluctuations on the Pacific coast using a sinusoidal function. Let \(H(t)\) give the height (in feet) above and below the average level of the ocean, where \(t\) is time in hours. Let \(t=0\) correspond to high tide. (b) Model the tidal fluctuations in the Bay of Fundy using a sinusoidal function. Use the same conventions as in part (a).

The average rental price for a two-bedroom apartment in Malden was \(\$ 800\) in 1990 and was \(\$ 1000\) in 2000 . The price has been increasing over the past decade. We want to model the price of a two-bedroom apartment in Malden as a function of time and use our model to predict the price in the year 2020 . Alex thinks that prices are increasing at a constant rate, so he models the price with a linear function, \(L(t) .\) Jamey thinks that the percent change in price is constant, so he models the price with an exponential function, \(E(t)\). Mike, an optimist who loves trigonometry, thinks that price is a sinusoidal function of time. He thinks that \(\$ 800\) is an all-time low and \(\$ 1000\) is an all-time high. He models the price with a sine or cosine function, \(T(t)\). (a) Suppose we let \(t=0\) correspond to the year 1990 and measure time in years. Find a formula for each of the following. Accompany your formula with a sketch. i. \(L(t)\) ii. \(E(t)\) iii. \(T(t)\) (b) Which model predicts the highest price for the year \(2003 ?\) Which model predicts the lowest price for the year \(2003 ?\) (c) What prices will Alex, Jamey, and Mike predict for the year \(2020 ?\) (d) Alex, Jamey, and Mike are combing the newspapers for information that might lend credence to one model over the other two. Which model, the linear, exponential, or trigonometric, is best supported by each of the following statements? i. "Prices in Malden have been growing at an increasing rate over the past decade." ii. "In the early 1990 s prices in Malden were increasing at an increasing rate. After 1995 the rate of increase began to drop off." iii. "Prices of apartments in Malden have been increasing very steadily over the past decade."

Determine whether or not each function is periodic. If a function is periodic, determine its period. (a) \(f(x)=\cos |x|\) (b) \(g(x)=\sin |x|\) (c) \(h(x)=|\cos x|\) (d) \(j(x)=|\sin x|\) (c) \(k(x)=\sin \left(x^{2}\right)\) (f) \(l(x)=\sin ^{2} x\)

Graph \(f(x)=\frac{1}{\cos x}\) on \([-\pi, 2 \pi]\).

Let \(f(x)=\frac{\sin x}{x}\). This function will be quite important when we are interested in the derivative of sine and cosine. (a) What is the domain of \(f(x)\) ? (b) Use a graphing calculator or computer to help you sketch the graph of \(f(x)\). (c) Although \(f(x)\) is undefined at \(x=0, \lim _{x \rightarrow 0} f(x)\) exists. What do you think this limit might be? Check out your conjecture numerically. Observe that if \(\lim _{x \rightarrow 0} \frac{\sin x}{x}=L\), then for \(x\) very close to zero, \(\frac{\sin x}{x} \approx L\), or, equivalently, \(\sin x \approx L x\) for \(x\) close to zero.

A typical person might have a pulse of 70 heartbeats per minute and a blood pressure reading of 120 over 80, where 120 is the high pressure and 80 is the low. Model blood pressure as a function of time using a sinusoidal function \(B(t)\), where \(t\) is time in minutes. (a) What is the amplitude of \(B(t)\) ? (b) What is the period of \(B(t) ?\) Notice that you have been given the frequency and from this must find the period. (c) Write a possible formula for \(B(t)\).