Trigonometric Functions

An object is traveling around a circle with a radius of $5$centimeters. If in $20$seconds a central angle of $\frac{1}{3}$radian is swept out, what is the angular speed of the object? What is its linear speed? 

An object is traveling around a circle with a radius of $2$ meters. If in $20$seconds the object travels $5$ meters, what is its angular speed? What is its linear speed? 

The diameter of each wheel of a bicycle is $26$ inches. If you are traveling at a speed of $35$ miles per hour on this bicycle, through how many revolutions per minute are the wheels turning? 

The radius of each wheel of a car is $15$ inches. If the wheels are turning at the rate of $3$ revolutions per second, how fast is the car moving? Express your answer in inches per second and in miles per hour. 

Memphis, Tennessee, is due north of New Orleans, Louisiana. Find the distance between Memphis (${35}^{°}{9}^{\text{'}}$ north latitude) and New Orleans (${29}^{°}{57}^{\text{'}}$north latitude). Assume that the radius of Earth is $3960$ miles.

Charleston, West Virginia, is due north of Jacksonville, Florida. Find the distance between Charleston (${38}^{\circ }21\text{'}$ north latitude) and Jacksonville (${30}^{\circ }20\text{'}$ north latitude). Assume that the radius of Earth is $3960$ miles. 

Linear Speed on Earth Earth rotates on an axis through its poles. The distance from the axis to a location on Earth 30° north latitude is about 3429.5 miles. Therefore, a location on Earth at 30° north latitude is spinning on a circle of radius of 3429.5 miles. Compute the linear speed on the surface of Earth at 30° north latitude.

Linear Speed on Earth Earth rotates on an axis through its poles. The distance from the axis to a location on Earth at  40° north latitude is about 3033.5 miles. Therefore, a location on Earth at 40° north latitude is spinning on a circle of radius of 3033.5 miles. Compute the linear speed on the surface of Earth at 40° north latitude.

The mean distance of the moon from Earth is $2.39 \times {10}^5$ miles. Assuming that the orbit of the moon around Earth is circular and that 1 revolution takes 27.3 days, find the linear speed of the moon. Express your answer in miles per hour. 

The mean distance of Earth from the Sun is $9.29 \times {10}^7$ miles. Assuming that the orbit of Earth around the Sun is circular and that 1 revolution takes 365 days, find the linear speed of Earth. Express your answer in miles per hour .

Two pulleys, one with radius 2 inches and the other with radius 8 inches, are connected by a belt. (See the figure.) If the 2-inch pulley is caused to rotate at 3 revolutions per minute, determine the revolutions per minute of the 8-inch pulley. 

A neighborhood carnival has a Ferris wheel whose radius is 30 feet. You measure the time it takes for one revolution to be 70 seconds. What is the linear speed (in feet per second) of this Ferris wheel? What is the angular speed in radians per second? 

To approximate the speed of the current of a river, a circular paddle wheel with radius 4 feet is lowered into the water. If the current causes the wheel to rotate at a speed of 10 revolutions per minute, what is the speed of the current? Express your answer in miles per hour. 

A spin balancer rotates the wheel of a car at 480 revolutions per minute. If the diameter of the wheel is 26 inches, what road speed is being tested? Express your answer in miles per hour. At how many revolutions per minute should the balancer be set to test a road speed of 80 miles per hour? 

The Cable Cars of San Francisco At the Cable Car Museum you can see the four cable lines that are used to pull cable cars up and down the hills of San Francisco. Each cable travels at a speed of 9.55 miles per hour, caused by a rotating wheel whose diameter is 8.5 feet. How fast is the wheel rotating? Express your answer in revolutions per minute.

Difference in Time of Sunrise Naples, Florida, is approximately 90 miles due west of Ft. Lauderdale. How much sooner would a person in Ft. Lauderdale first see the rising Sun than a person in Naples? See the hint. [Hint: Consult the figure. When a person at Q sees the first rays of the Sun, a person at P is still in the dark. The person at P sees the first rays after Earth has rotated so that P is at the location Q. Now use the fact that at the latitude of Ft. Lauderdale in 24 hours an arc of length $2\mathrm{\pi }\left(3559\right)$ miles is subtended.]

Keeping Up with the Sun How fast would you have to travel on the surface of Earth at the equator to keep up with the Sun (that is, so that the Sun would appear to remain in

the same position in the sky)?

Nautical Miles A nautical mile equals the length of arc subtended by a central angle of 1 minute on a great circle† on the surface of Earth. See the figure. If the radius of Earth is taken as 3960 miles, express 1 nautical mile in terms of ordinary, or statute, miles.

Approximating the Circumference of Earth Eratosthenes of Cyrene (276–195 bc) was a Greek scholar who lived and worked in Cyrene and Alexandria. One day while visiting in

Syene he noticed that the Sun’s rays shone directly down a well. On this date 1 year later, in Alexandria, which is 500 miles due north of Syene he measured the angle of the Sun to be about 6.2 degrees. See the figure. Use this information to approximate the radius and circumference of Earth.

Designing a Little League Field For a 60-foot Little League Baseball field, the distance from home base to the nearest fence (or other obstruction) in fair territory should

be a minimum of 200 feet. The commissioner of parks and recreation is making plans for a new 60-foot field. Because of limited ground availability, he will use the minimum required distance to the outfield fence. To increase safety, however, he plans to include a 10-foot wide warning track on the inside of the fence. To further increase safety, the fence and warning track will extend both directions into foul territory. In total, the arc formed by the outfield fence (including the extensions into the foul territories) will be subtended by a central angle at home plate measuring 96°, as illustrated.

(a) Determine the length of the outfield fence.

(b) Determine the area of the warning track.

Pulleys Two pulleys, one with radius $r_1$ and the other with radius $r_2$, are connected by a belt. The pulley with radius $r_1$rotates at $v_1$revolutions per minute, whereas the pulley with

radius $r_2$ rotates at $v_2$ revolutions per minute. Show that $\frac{r_1}{r_2}=\frac{{\omega }_1}{{\omega }_2}$

Do you prefer to measure angles using degrees or radians? Provide justification and a rationale for your choice.

What is 1 radian? What is 1 degree?

Which angle has the larger measure: 1 degree or 1 radian? Or are they equal?

Explain the difference between linear speed and angular speed.

For a circle of radius r, a central angle of u degrees subtends an arc whose length s is$s=\frac{\mathrm{\pi }}{180}r\theta$  . Discuss whether this is a true or false statement. Give reasons to defend your position.

Discuss why ships and airplanes use nautical miles to measure distance. Explain the difference between a nautical mile and a statute mile.

Investigate the way that speed bicycles work. In particular, explain the differences and similarities between 5-speed and 9-speed derailleurs. Be sure to include a discussion of linear

speed and angular speed.

In Example 6, we found that the distance between Albuquerque, New Mexico, and Glasgow, Montana, is approximately 903 miles. According to mapquest.com, the distance is approximately 1300 miles. What might account for the difference?

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

In a right triangle, with legs$a$ and $b$and hypotenuse$c$, the Pythagorean Theorem states that _________.

The value of the function $f\left(x\right)=3x-7$ at 5 is ________.

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

True or False For a function $y=f\left(x\right)$, for each $x$ in the domain, there is exactly one element$y$ in the range.

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

If two triangles are similar, then corresponding angles are ______ and the lengths of corresponding sides are ______.

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

What point is symmetric with respect to the $y-$axis to the point $\left(\frac{1}{2},\frac{\sqrt{3}}{2}\right)$.

Answers are given at the end of these exercises. If you get a wrong answer, read the pages listed in red.

If $\left(x,y\right)$ is a point on the unit circle in quadrant $\mathrm{IV}$ and if $x=\frac{\sqrt{3}}{2}$ what is $y$?

The__________ function takes as input a real number $t$that corresponds to a point$P=\left(x,y\right)$ on the unit circle and outputs the $x$-coordinate.

The point on the unit circle that corresponds to $\theta =\frac{\pi }{2}$ is $P=$_________.

The point on the unit circle that corresponds to $\theta =\frac{\pi }{4}$ is $P=$_____________.

The point on the unit circle that corresponds to $\theta =\frac{\pi }{3}$ is $P=$________.

For any angle $\theta$ in standard position, let $P=\left(x,y\right)$ be the point on the terminal side of $\theta$ that is also on the circle$x^2+y^2=r^2$.

 Then, $\sin \theta$ =________ and $\cos \theta$ = ________.

Exact values can be found for the sine of any angle.

In Problems 13–20, $P=\left(x,y\right)$ is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(\frac{\sqrt{3}}{2},\frac{1}{2}\right)$.

In Problems 13–20, $P\left(x,y\right)$is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(\frac{1}{2},-\frac{\sqrt{3}}{2}\right)$.

In Problems 13–20, $P\left(x,y\right)$is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(-\frac{2}{5},\frac{\sqrt{21}}{5}\right)$.

In problems 13-20, $P\left(x,y\right)$ is the  point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(-\frac{1}{5},\frac{2\sqrt{6}}{5}\right)$.

In problems 13-20, $P\left(x,y\right)$ is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(-\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$.

In problems 13-20, $P\left(x,y\right)$ is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(\frac{\sqrt{2}}{2},\frac{\sqrt{2}}{2}\right)$.

In problems 13-20, $P\left(x,y\right)$ is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(\frac{2\sqrt{2}}{3},-\frac{1}{3}\right)$.

In problems 13-20, $P\left(x,y\right)$ is the point on the unit circle that corresponds to a real number $t$. Find the exact values of the six trigonometric functions of $t$.

$\left(-\frac{\sqrt{5}}{3},-\frac{2}{3}\right)$.

Find the exact value. Do not use a calculator.

$\sin \frac{11\mathrm{\pi }}{2}$