Chapter 5

Find the sine and cosine of the angle \(z\) in \([0,2 \pi),\) in standard position, cohose terminal side intersects the unit circle at the giecn point. $$\left(-\frac{4}{5}, \frac{3}{5}\right)$$

A bicycle with tires of 18 -inch radius travels at a speed of 20 mph. What is the angular speed of the tires? Express your answer in both degrees per second and radians per second.

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A garden consists of four sections of equal dimensions. Each section is in the shape of a right triangle. One of the acute angles in each triangle measures \(24^{\circ},\) and the side adjacent to that angle is 16 feet long. (IMAGES CANNOT COPY) (a) Find the length of the hypotenuse. (b) Find the length of the side opposite the \(24^{\circ}\) angle. (c) Compute the total area of the garden (all four sections combined).

Find the sine and cosine of the angle \(z\) in \([0,2 \pi),\) in standard position, cohose terminal side intersects the unit circle at the giecn point. $$(0.8,-0.6)$$

A circular music box rotates at a constant rate while the music is playing. What is the linear speed of a fly that is perched on the music box at a point 2 inches from its center if it takes the music box 6 seconds to make one revolution? Express your answer in inches per second.

Find the sine and cosine of the angle \(z\) in \([0,2 \pi),\) in standard position, cohose terminal side intersects the unit circle at the giecn point. $$\left(-\frac{12}{13},-\frac{5}{13}\right)$$

In carly 2007 , the singer Justin Timberlake toured "in the round." This means that the stage was round and rotated during the performance. If the stage made one complete rotation every 10 minutes and Mr. Timberlake stood at a distance of 20 feet from the center of the stage during one of the songs, what was his linear speed in feet per minute?

Find the sine and cosine of the angle \(z\) in \([0,2 \pi),\) in standard position, cohose terminal side intersects the unit circle at the giecn point. $$\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$$

A weight is moved upward through the use of a pulley 10 inches in radius. If the pulley is rotated counterclockwise through an angle of 45 ", approximate the height, in inches, that the weight will rise. Round your answer to two decimal places.

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\cos s=\cos t\) $$t=\frac{\pi}{4}$$

In this set of exercises, you will use right triangle trigonometry to study real-world problems. Unless otherwise indicated, round answers to four decimal places. A photograph is 8 inches wide and 10 inches high. It must be trimmed for publication in a magazine. A 3 -inch-wide horizontal strip is to be trimmed from the top and a vertical strip is cut from the right. Let \(x\) be the width of the vertical strip that is to be trimmed from the right side of the photo. What is the value of \(x\) if the point that is to be in the upper right corner of the trimmed photograph lies along the line that makes an angle of \(60^{\circ}\) with the top edge of the original (untrimmed) photograph? The vertex of the \(60^{\circ}\) angle is at the upper-right corner of the original photo.

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\cos s=\cos t\) $$t=\frac{\pi}{2}$$

A scamstress secures one end of a piece of thread to a spool. Then she uses an attachment on her scwing machine to cause the spool to spin around, which in turn causes the thread to wind around the spool. If the spool has a diameter of \(1.6 \mathrm{cm}\) and spins at a rate of 3 revolutions per second, what length of thread (in centimeters) is wound around the spool in 1 second?

This set of exercises will draw on the ideas presented in this section and your general math background. $$\text { Show that } \tan \left(90^{\circ}-\theta\right)=\cot \theta$$

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\cos s=\cos t\) $$t=\frac{4 \pi}{3}$$

In one of the rides at an amusement park, you sit in a circular "car" and cause it to rotate by turning a wheel in the center. The faster you turn the whecl, the faster the car rotates. How far from the center of the car are you sitting if your car makes one revolution cuery 3 seconds and your lincar speed is 5 feet per second? Express your answer in feet.

This set of exercises will draw on the ideas presented in this section and your general math background. $$\text { Show that } \csc \left(90^{\circ}-\theta\right)=\sec \theta$$

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\cos s=\cos t\) $$t=\frac{11 \pi}{6}$$

This set of exercises will draw on the ideas presented in this section and your general math background. Can a right triangle be used to define \(\sin 90^{\circ} ?\)

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\sin s=\sin t\) $$t=\pi$$

A game played by many children involves placing a cuff around one ankle that has a ball attached to it by a string 2 feet long. The ball is spun around the child's leg while he or she jumps over the rope with the other foot. Suppose the ball is making one revolution per second. Calculate the linear speed of the ball in feet per second.

This set of exercises will draw on the ideas presented in this section and your general math background. Before the widespread use of calculators, values of the sine and cosine for selected angles in the interval \(\left[0^{\circ}, 45^{\circ}\right]\) were given in tables. Why was there no need to list the sine and cosine for angles in the interval \(\left(45^{\circ}, 90^{\circ}\right) ?\)

The first ferris wheel was 250 feet in diameter. It was invented by John Ferris in \(1893 .\) Assuming it made one revolution every 30 seconds, what was the angular speed of a passenger (assume the passenger is on the edge of the wheel) in degrees per minute? What was the passenger's linear speed in feet per minute?

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\sin s=\sin t\) $$t=\frac{2 \pi}{3}$$

Earth rotates about an axis through its poles, making one revolution per day. (a) What is the exact angular speed of Earth about its axis? Express your answer in both degrees per hour and radians per hour. (b) The radius of Earth is approximately 3900 miles. What distance is traversed by a point on Earth's surface at the equator during any 8 -hour interval as a result of Earth's rotation about its axis? Express your answer in miles. (c) What is the linear speed (in miles per hour) of the point in part (b)?

Find an angle s such that \(s \neq t, 0 \leq s<2 \pi\) and \(\sin s=\sin t\) $$t=\frac{3 \pi}{4}$$

Consider an angle \(\theta\) in standard position whose vertex coincides with the center of a circle of radius \(r .\) The portion of the circle bounded by the initial side and the terminal side of the angle \(\theta\) is called a sector of the circle. (a) If \(A\) is the area of the circle, then \(A_{s}=A \frac{\theta}{2 \pi}\) represents the area of the sector because \(\frac{\theta}{2 \pi}\) gives the fraction of the area covered by the sector. Show that the area of a sector, \(A_{s},\) is \(A_{s}=\frac{r^{2} \theta}{2} .\) Here theta is in radians. (b) Find \(A_{s}\) if \(\theta=\frac{\pi}{3}\) and \(r=12\) inches.

Find exact values of \(\cos 3 t\) and \(\cos \left(\frac{t}{3}\right)\) for the given values of \(t\) $$t=0$$

Which is the larger angle, \(1^{\circ}\) or 1 radian? Explain.

Find exact values of \(\cos 3 t\) and \(\cos \left(\frac{t}{3}\right)\) for the given values of \(t\) $$t=\frac{\pi}{2}$$

What is the area of the portion of the unit circle swept Sut by an angle of \(\frac{\pi}{6}\) radians?

Find the radian measure of an angle in standard position that is generated by the specified rotation. Quarter of a full revolution clockwise

Find exact values of \(\cos 3 t\) and \(\cos \left(\frac{t}{3}\right)\) for the given values of \(t\) $$t=-\frac{\pi}{2}$$

Find the radian measure of an angle in standard position that is generated by the specified rotation. Half of a full revolution counterclockwise

Find exact values of \(\cos 3 t\) and \(\cos \left(\frac{t}{3}\right)\) for the given values of \(t\) $$t=-\pi$$

Find the radian measure of an angle in standard position that is generated by the specified rotation. One-third of a full revolution counterclockwise

This set of exercises will draw on the ideas presented in this section and your general math background. Does the equation \(\sin (t+\pi)=\sin t+\sin \pi\) hold for all \(t\) ? Explain.

Find the radian measure of an angle in standard position that is generated by the specified rotation. Two-thirds of a full revolution clockwise

This set of exercises will draw on the ideas presented in this section and your general math background. Does the equation \(\cos \left(\frac{t}{2}\right)=\frac{\cos t}{2}\) hold for all \(t\) ? Explain.

Find the radian measure of an angle in standard position that is generated by the specified rotation. Two full revolutions clockwise

This set of exercises will draw on the ideas presented in this section and your general math background. Find all values of \(t\) in \([0,2 \pi)\) such that \(\sin t=\cos t\)

Find the radian measure of an angle in standard position that is generated by the specified rotation. Three full revolutions counterclockwise

Find all values of \(t\) in \([0,2 \pi)\) such that \(\cos t=-\frac{1}{2}\)

Suppose \(t\) is in \(\left(0, \frac{\pi}{2}\right) .\) Express \(\cos (t+\pi)\) in terms of cos \(t .\) (Hint: It is helpful to sketch a figure.)

Suppose \(t\) is in \(\left(0, \frac{\pi}{2}\right) .\) Express \(\sin \left(t+\frac{\pi}{2}\right)\) in terms of sin \(t .\) (Hint: It is helpful to sketch a figure.)

Show that the points \((\cos t, \sin t)\) and \(\left(\cos \left(\frac{\pi}{2}-t\right)\right.\) \(\left.\sin \left(\frac{\pi}{2}-t\right)\right)\) are symmetric with respect to the line \(y=x\) for \(t\) in \(\left[0, \frac{\pi}{4}\right)\)

Derive the Pythagorean identity \(1+\cot ^{2} t=\csc ^{2} t\)