Convert the following degree measures to radians. Leave answers as multiples of $\pi$.

$$

65^{\circ}

$$

Sherrie F.

Numerade Educator

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$

$$90^{\circ}$$

Sid W.

University of Louisville

Convert the following degree measures to radians. Leave answers as multiples of $\pi$.

$$

95^{\circ}

$$

Sherrie F.

Numerade Educator

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$

$$135^{\circ}$$

Sid W.

University of Louisville

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$

$$270^{\circ}$$

Sherrie F.

Numerade Educator

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$

$$320^{\circ}$$

Sid W.

University of Louisville

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$

$$495^{\circ}$$

Joel N.

Numerade Educator

Convert the following degree measures to radians. Leave answers as multiples of $\pi .$

$$510^{\circ}$$

Sid W.

University of Louisville

Convert the following radian measures to degrees.

$$\frac{7 \pi}{6}$$

Sid W.

University of Louisville

Convert the following radian measures to degrees.

$$-\frac{13 \pi}{6}$$

Nathan C.

Numerade Educator

Convert the following radian measures to degrees.

$$-\frac{9 \pi}{4}$$

Sid W.

University of Louisville

Convert the following radian measures to degrees.

$$\frac{8 \pi}{5}$$

Sherrie F.

Numerade Educator

Convert the following radian measures to degrees.

$$\frac{9 \pi}{5}$$

Sid W.

University of Louisville

Convert the following radian measures to degrees.

$$\frac{7 \pi}{12}$$

Kanika A.

Numerade Educator

Convert the following radian measures to degrees.

$$5 \pi$$

Sid W.

University of Louisville

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.

$$(-3,4)$$

Sherrie F.

Numerade Educator

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.

$$(-12,-5)$$

Sid W.

University of Louisville

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.

$$(7,-24)$$

Sherrie F.

Numerade Educator

Find the values of the six trigonometric functions for the angles in standard position having the points in Exercises $17-20$ on their terminal sides.

$$(20,15)$$

Sid W.

University of Louisville

In quadrant $I, x, y,$ and $r$ are all positive, so that all six trigonometric functions have positive values. In quadrant II, $x$ is negative and $y$ is positive $(r$ is always positive). Thus, in quadrant II, sine is positive, cosine is negative, and so on. For Exercises $21-24,$ complete the following table of values for the signs of the trigonometric functions.

Sherrie F.

Numerade Educator

Sid W.

University of Louisville

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Sid W.

University of Louisville

For Exercises $25-32,$ complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$30^{\circ} \quad 1 / 2 \quad \sqrt{3} / 2 \quad \sqrt{3} / 2$

Sherrie F.

Numerade Educator

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$45^{\circ}$

Sid W.

University of Louisville

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$60^{\circ}$

Sherrie F.

Numerade Educator

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$120^{\circ}$

Sid W.

University of Louisville

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$135^{\circ}$

Sherrie F.

Numerade Educator

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$150^{\circ}$

Sid W.

University of Louisville

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$210^{\circ}$

Sherrie F.

Numerade Educator

For Exercises $25-32$ , complete the following table. Use the $30^{\circ}-60^{\circ}-90^{\circ}$ and $45^{\circ}-45^{\circ}-90^{\circ}$ triangles. Do not use a calculator.

$240^{\circ}$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\sin \frac{\pi}{3}$$

Sherrie F.

Numerade Educator

Find the following function values without using a calculator.

$$\cos \frac{\pi}{6}$$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\tan \frac{\pi}{4}$$

Sherrie F.

Numerade Educator

Find the following function values without using a calculator.

$$\cot \frac{\pi}{3}$$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\csc \frac{\pi}{6}$$

Jonathan P.

Numerade Educator

Find the following function values without using a calculator.

$$\sin \frac{3 \pi}{2}$$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\cos 3 \pi$$

Sherrie F.

Numerade Educator

Find the following function values without using a calculator.

$$\sec \pi$$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\sin \frac{7 \pi}{4}$$

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Find the following function values without using a calculator.

$$\tan \frac{5 \pi}{2}$$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\sec \frac{5 \pi}{4}$$

Sherrie F.

Numerade Educator

Find the following function values without using a calculator.

$$\cos 5 \pi$$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\cot -\frac{3 \pi}{4}$$

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Find the following function values without using a calculator.

$$\tan -\frac{5 \pi}{6}$$

Sid W.

University of Louisville

Find the following function values without using a calculator.

$$\sin -\frac{7 \pi}{6}$$

Sherrie F.

Numerade Educator

Find the following function values without using a calculator.

$$\cos -\frac{\pi}{6}$$

Sid W.

University of Louisville

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.

$$\cos \theta=1 / 2$$

Sherrie F.

Numerade Educator

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.

$$\sin \theta=-1 / 2$$

Sid W.

University of Louisville

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.

$$\tan \theta=-1$$

Sherrie F.

Numerade Educator

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.

$$\tan \theta=\sqrt{3}$$

Sid W.

University of Louisville

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.

$$\sec \theta=-2 / \sqrt{3}$$

Sherrie F.

Numerade Educator

Find all values of $\theta$ between 0 and 2$\pi$ that satisfy each of the following equations.

$$\sec \theta=\sqrt{2}$$

Sid W.

University of Louisville

Use a calculator to find the following function values.

$$\sin 39^{\circ}$$

Christine A.

Numerade Educator

Use a calculator to find the following function values.

$$\cos 67^{\circ}$$

Sid W.

University of Louisville

Use a calculator to find the following function values.

$$\tan 123^{\circ}$$

Sherrie F.

Numerade Educator

Use a calculator to find the following function values.

$$\tan 54^{\circ}$$

Sid W.

University of Louisville

Use a calculator to find the following function values.

$$\sin 0.3638$$

Sherrie F.

Numerade Educator

Use a calculator to find the following function values.

$$\tan 1.0123$$

Sid W.

University of Louisville

Use a calculator to find the following function values.

$$\cos 1.2353$$

Sherrie F.

Numerade Educator

Use a calculator to find the following function values.

$$\sin 1.5359$$

Sid W.

University of Louisville

Find the amplitude $(a)$ and period $(T)$ of each function.

$$f(x)=\cos (3 x)$$

Sherrie F.

Numerade Educator

Find the amplitude $(a)$ and period $(T)$ of each function.

$$h(x)=-\frac{1}{2} \sin (4 \pi x)$$

Sid W.

University of Louisville

Find the amplitude $(a)$ and period $(T)$ of each function.

$$g(t)=-2 \sin \left(\frac{\pi}{4} t+2\right)$$

Sherrie F.

Numerade Educator

Find the amplitude $(a)$ and period $(T)$ of each function.

$$s(t)=3 \sin (880 \pi t-7)$$

Sid W.

University of Louisville

Graph each function defined as follows over a two-period interval.

$$y=2 \cos x$$

Sherrie F.

Numerade Educator

Graph each function defined as follows over a two-period interval.

$$y=2 \sin x$$

Sid W.

University of Louisville

Graph each function defined as follows over a two-period interval.

$$y=-\frac{1}{2} \cos x$$

Sherrie F.

Numerade Educator

Graph each function defined as follows over a two-period interval.

$$y=-\sin x$$

Sid W.

University of Louisville

Graph each function defined as follows over a two-period interval.

$$y=4 \sin \left(\frac{1}{2} x+\pi\right)+2$$

Sherrie F.

Numerade Educator

Graph each function defined as follows over a two-period interval.

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

Sid W.

University of Louisville

Graph each function defined as follows over a two-period interval.

$$y=-3 \tan x$$

Kiska Z.

Numerade Educator

Graph each function defined as follows over a two-period interval.

$$y=\frac{1}{2} \tan x$$

Sid W.

University of Louisville

Consider the triangle shown on the next page, in which the three angles $\theta$ are equal and all sides have length $2 .$

(a) Using the fact that the sum of the angles in a triangle is $180^{\circ}$ , what are the measures of the three equal angles $\theta$ ?

(b) Suppose the triangle is cut in half as shown by a vertical line. What are the measures of the angles in the blue tri- angle on the left?

(c) What are the measures of the sides of the blue triangle on the left? (Hint: Once you've found the length of

Sherrie F.

Numerade Educator

Consider the right triangle shown, in which the two sides have length $1 .$

(a) Using the Pythagorean Theorem, what is the length of the hypotenuse?

(b) Using the fact that the sum of the angles in a triangle is $180^{\circ}$ , what are the measures of the three angles?

Sid W.

University of Louisville

Sales Sales of snowblowers are seasonal. Suppose the sales of snowblowers in one region of the country are approximated by

$$ S(t)=500+500 \cos \left(\frac{\pi}{6} t\right) $$

where $t$ is time in months, with $t=0$ corresponding to November. Find the sales for $(a)-(e)$

(a) November $\quad$ (b) January $\quad$ (c) February (d) May (e) August $\quad$ (f) Graph $y=S(t)$

Sherrie F.

Numerade Educator

APPLY IT Electricity Consumption The amount of electricity (in trillion BTUs) consumed by U.S. residential customers in 2013 is given in the following table. Source: Energy Information Administration.

(a) Plot the data, letting $t=1$ correspond to January, $t=2$ to February, and so on. Is it reasonable to assume that electri- cal consumption is periodic?

(b) Use a calculator with trigonometric regression to find a trigonometric function of the form

$$C(t)=a \sin (b t+c)+d$$

that models these data when $t$ is the month and $C(t)$ is the amount of electricity consumed (in trillion BTUs). Graph the function on the same calculator window as the data.

(c) Determine the period, $T,$ of the function found in part (b). Discuss the reasonableness of this period.

(d) Use the function from part (b) to estimate the consumption for the month of October, and compare it to the actual value.

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Transylvania Hypothesis The "Transylvania hypothesis" claims that the full moon has an effect on health-related behavior. A study investigating this effect found a significant relationship between the phase of the moon and the number of general practice consultations nationwide, given by

$$y=100+1.8 \cos \left[\frac{(t-6) \pi}{14.77}\right],$$

where $y$ is the number of consultations as a percentage of the daily mean and $t$ is the days since the last full moon. Source: Family Practice.

(a) What is the period of this function? What is the significance of this period?

(b) There was a full moon on October $8,2014 .$ On what day in October 2014 does this formula predict the maximum number of consultations? What percent increase would be predicted for that day?

(c) What does the formula predict for October $25,2014 ?$

Sherrie F.

Numerade Educator

Hyperkalemia Diagnosis A person with Hyperkalemia is monitored on an ECG (electrocardiogram). The graph shows that he has an inter-beat interval (conventionally named RR interval) of 32 seconds, an amplitude of 1.23 volts, a vertical shift of 2.4 volts, and a horizontal shift of 5 seconds.

(a) Find an equation giving the voltage of electricity as a function of time in seconds.

(b) After how many seconds does the electricity reaches its maximum amplitude?

(c) What is the value of voltage after 2 minutes?

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Alzheimer's Disease A study on the circadian rhythms of patients with Alzheimer's disease found the body temperature of patients could be described by a function of the form

$$T=T_{0}+a \cos \left(\frac{2 \pi(t-k)}{24}\right),$$

where $t$ is the time in hours since midnight. For the patients without Alzheimer's, the average values of $T_{0}$ (the MESOR), $a$ (the amplitude), and $k$ (the acrophase) were $36.91^{\circ} \mathrm{C}, 0.32^{\circ} \mathrm{C},$ and 14.92 hours, while for the patients with the disease, the values were $37.29^{\circ} \mathrm{C}$ $0.46^{\circ} \mathrm{C},$ and 16.37 hours. Source: American Journal of Psychiatry.

(a) Graph the functions giving the temperature for each of the two groups using a graphing calculator. Do these two functions ever cross?

(b) At what time is the temperature highest for the patients without Alzheimer's?

(c) At what time is the temperature highest for the patients with Alzheimer's?

Sherrie F.

Numerade Educator

Air Pollution The amount of pollution in the air fluctuates with the seasons. It is lower after heavy spring rains and higher after periods of little rain. In addition to this seasonal fluctuation, the long-term trend in many areas is upward. An idealized graph of this situation is shown in the figure below. Trigonometric functions can be used to describe the fluctuating part of the pollution levels. Powers of the number e can be used to show the long-term growth. In fact, the pollution level in a

certain area might be given by

$$P(t)=7(1-\cos 2 \pi t)(t+10)+100 e^{0.2 t}$$

where $t$ is time in years, with $t=0$ representing January 1 of the base year. Thus, July 1 of the same year would be represented by $t=0.5,$ while October 1 of the following year would be represented by $t=1.75 .$ Find the pollution levels on the following dates.

$\begin{array}{ll}{\text { (a) January } 1, \text { base year }} & {\text { (b) July } 1, \text { base year }} \\ {\text { (c) January } 1, \text { following year }} & {\text { (d) July } 1, \text { following year }}\end{array}$

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Air Pollution Using a computer or a graphing calculator, sketch the function for air pollution given in Exercise 82 over the interval $[0,6] .$

Sherrie F.

Numerade Educator

In Exercises 84 and $85,$ assume that $c_{1}=3 \times 10^{8} m$ per second, and find the speed of light in the second medium.

$$\theta_{1}=39^{\circ}, \theta_{2}=28^{\circ}$$

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In Exercises 84 and $85,$ assume that $c_{1}=3 \times 10^{8} m$ per second, and find the speed of light in the second medium.

$$\theta_{1}=46^{\circ}, \theta_{2}=31^{\circ}$$

Sherrie F.

Numerade Educator

Sound Pure sounds produce single sine waves on an oscilloscope. Find the period of each sine wave in the photographs in Exercises 86 and $87 .$ On the vertical scale each square represents $0.5,$ and on the horizontal scale each square represents $30^{\circ} .$

Sid W.

University of Louisville

Sherrie F.

Numerade Educator

Sound Suppose the A key above Middle $\mathrm{C}$ is played as a pure tone. For this tone,

$$P(t)=0.002 \sin (880 \pi t)$$

where $P(t)$ is the change of pressure (in pounds per square foot) on a person's eardrum at time $t$ (in seconds). Source: The Physics and Psychophysics of Music: An Introduction.

(a) Graph this function on $[0,0.003]$ .

(b) Determine analytically the values of $t$ for which $P=0$ on $[0,0.003]$ and check graphically.

(c) Determine the period $T$ of $P(t)$ and the frequency of the A note.

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Temperature The maximum afternoon temperature degrees Fahrenheit) in a given city is approximated by

$$T(t)=60-30 \cos (t / 2)$$

where $t$ represents the month, with $t=0$ representing January, $t=1$ representing February, and so on. Use a calculator to find the maximum afternoon temperature for the following months.

(a) Februry $\quad$ (b) April $\quad$ (c) September (d) July (e) December

Sherrie F.

Numerade Educator

Temperature $\quad$ A mathematical model for the temperature in Fairbanks is

$$T(t)=37 \sin \left[\frac{2 \pi}{365}(t-101)\right]+25$$

where $T(t)$ is the temperature (in degrees Fahrenheit) on day $t$ with $t=0$ corresponding to January 1 and $t=364$ corresponding to December $31 .$ Use a calculator to estimate the tem- perature for $(a)-(d) .$ Source: The Mathematics Teacher.

(a) March 16$($ Day 74$)$ (b) May 2 (Day 121$)$ (c) Day 250 (d) Day 325 (e) Find maximum and minimum values of $T .$ (f) Find the period, T.

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Sunset The number of minutes after noon, Eastern Standard Time, that the sun sets in Boston for specific days of the year is approximated in the following table. Source: The Old Farmer's Almanac.

(a) Plot the data. Is it reasonable to assume that the times of sunset are periodic?

(b) Use a calculator with trigonometric regression to find a trigonometric function of the form $s(t)=a \sin (b t+c)+d$ that models these data when $t$ is the day of the year and $s(t)$ is the number of minutes past noon, Eastern Standard Time that the sun sets

(c) Estimate the time of sunset for days $60,120,240 .$ Round answers to the nearest minute. (Hint: Don't forget about daylight savings time.)

(d) Use part (b) to estimate the days of the year that the sun sets at $6 : 00$ p.M. In reality, the days are close to 82 and $290 .$

Sherrie F.

Numerade Educator

Cameras In the Kodak Customer Service Pamphlet $A A-26$ , Optical Formulas and Their Applications, the near and far limits of the depth of field (how close or how far away an object can be placed and still be in focus) are given by

$$w_{1}=\frac{u^{2}(\tan \theta)}{L+u(\tan \theta)} \quad$ and $\quad \mathrm{w}_{2}=\frac{u^{2}(\tan \theta)}{L-u(\tan \theta)}$$

In these equations, $\theta$ represents the angle between the lens and the "circle of confusion," which is the circular image on the film of a point that is not exactly in focus. (The pamphlet suggests letting $\theta=1 / 30^{\circ} .$ ) $L$ is the diameter of the lens opening, which is found by dividing the focal length by the f-stop. (This is camera jargon you need not worry about here.) For this problem, let the focal length be $50 \mathrm{mm},$ or $0.05 \mathrm{m} ;$ if the lens is set at $\mathrm{f} / 8,$ then $L=0.05 / 8=0.00625 \mathrm{m} .$ Finally, $u$ is the distance to the object being photographed. Find the near and far limits of the depth of field when the object being photographed is 6 $\mathrm{m}$ from the camera.

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Measurement A surveyor standing 65 $\mathrm{m}$ from the base of a building measures the angle to the top of the building and finds it to be $42.8^{\circ} .$ (See the figure.) Use trigonometry to find the height of the building.

Sherrie F.

Numerade Educator

Measurement Jenny Crum stands on a cliff at the edge of a canyon. On the opposite side of the canyon is another cliff equal in height to the one she is on. (See the figure.) By dropping a rock and timing its fall, she determines that it is 105 ft to the bottom of the canyon. She also determines that the angle to the base of the opposite cliff is $27^{\circ} .$ How far is it to the opposite side of the canyon?

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Whitewater Rafting A mathematics textbook author rafting down the Colorado River was told by a guide that the river dropped an average of 26 ft per mile as it ran through Cataract Canyon. Find the average angle of the river with the horizontal in degrees. Hint: Find the tangent of the angle, and then use a calculator to find the angle where the tangent has that value. There are 5280 ft in a mile. Be sure your calculator is set on degrees.)

Sherrie F.

Numerade Educator

Computer Drawing A mathematics professor wanted to use a computer drawing program to draw a picture of a regular pentagon (a five-sided figure with sides of equal length and with equal angles). He first made a 1 -in. base by drawing a line from $(0,0)$ to $(1,0) .$ (See the figure.) He then needed to find the coordinates of the other three vertex points. Use trigonometry to find them. (Hint: The sum of the exterior angles of any polygon is $360^{\circ} . )$

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Amusement Rides A proud father is attempting to take a picture of his daughters while they are riding on a merry-go-round. Horses on this particular ride move up and down as the ride progresses according to the function

$$h(t)=\sin \left(\frac{t}{\pi}-2\right)+4$$

where $h(t)$ represents the height (in feet) of the horse's nose at time $t,$ relative to the merry-go-round platform. However, because of safety fencing surrounding the ride, it is only possible to get a good picture when the height of the horse's nose is between 3.5 and 4 ft off the merry-go-round platform. Find the first time interval that the father has to take the picture.

Sherrie F.

Numerade Educator