Chapter 5

Use a calculator to evaluate the trigonometric function. Round your answer to four decimal places. (Be sure the calculator is set in the correct angle mode.) $$\cos (-15 \pi / 14)$$

Find the radius \(r\) of a circle with an arc length \(s\) and a central angle \(\theta\). Arc Length \(s\) 3 meters Central Angle \(\theta\) \(\frac{4 \pi}{3}\) radians

Plot the points and find the slope of the line passing through the points. $$(0,1),(2,5)$$

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\cos t$$

Find the radius \(r\) of a circle with an arc length \(s\) and a central angle \(\theta\). Arc Length \(s\) 82 miles Central Angle \(\theta\) \(135^{\circ}\)

Plot the points and find the slope of the line passing through the points. $$(-1,4),(3,-2)$$

Determine whether the statement is true or false. Justify your answer. $$\text { 1. } \tan \frac{5 \pi}{4}=1 \quad \square \quad \text { arctan } 1=\frac{5 \pi}{4}$$

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\sin t$$

Find the radius \(r\) of a circle with an arc length \(s\) and a central angle \(\theta\). Arc Length \(s\) 8 inches Central Angle \(\theta\) \(330^{\circ}\)

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$8.5$$

Determine whether the statement is true or false. Justify your answer. \(\arctan x=\frac{\arcsin x}{\arccos x}\)

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\tan t$$

Find the distance between the cities. Assume that Earth is a sphere of radius 4000 miles and the cities are on the same longitude (one city is due north of the other). City Dallas, Texas Omaha, Nebraska Latitude \(32^{\circ} 47^{\prime} 39^{\prime \prime} \mathrm{N}\) \(41^{\circ} 15^{\prime} 50^{\prime \prime} \mathrm{N}\)

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$-0.48$$

Evaluate the function for the given values of \(t\). Values of \(t\) (a) \(t=3\) (a) \(t=2\) (a) \(t=0.3\) (a) \(t=0.1\) (b) \(t=\pi\) (b) \(t=0\) (b) \(t=-\pi / 4\) (b) \(t=-2 \pi / 3\) Trigonometric Function $$f(t)=\cos t$$

Find the distance between the cities. Assume that Earth is a sphere of radius 4000 miles and the cities are on the same longitude (one city is due north of the other). City San Francisco, California Seattle, Washington Latitude \(37^{\circ} 47^{\prime} 36^{\prime \prime} \mathrm{N}\) \(47^{\circ} 37^{\prime} 18^{\prime \prime} \mathrm{N}\)

Define the inverse cotangent function by restricting the domain of the cotangent function to the interval \((0, \pi),\) and sketch the graph of the inverse function.

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\sin \theta=\frac{1}{2}\) (b) \(\sin \theta=-\frac{1}{2}\)

Assuming that Earth is a sphere of radius 6378 kilometers, what is the difference in the latitudes of Syracuse, New York, and Annapolis, Maryland, where Syracuse is 450 kilometers due north of Annapolis?

Define the inverse secant function by restricting the domain of the secant function to the intervals \([0, \pi / 2)\) and \((\pi / 2, \pi],\) and sketch the graph of the inverse function.

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\cos \theta=\frac{\sqrt{2}}{2}\) (b) \(\cos \theta=-\frac{\sqrt{2}}{2}\)

Define the inverse cosecant function by restricting the domain of the cosecant function to the intervals \([-\pi / 2,0)\) and \((0, \pi / 2],\) and sketch the graph of the inverse function.

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\csc \theta=\frac{2 \sqrt{3}}{3}\) (b) \(\cot \theta=-1\)

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\cot \theta=-\sqrt{3}\) (b) \(\csc \theta=2\)

The number ofrevolutions made by a figure skater for each type of axel jump is given. Determine the measure of the angle generated as the skater performs each jump. Give the answer in both degrees and radians. (a) single axel: \(1 \frac{1}{2}\) revolutions (b) Double axel: \(2 \frac{1}{2}\) revolutions (c) Triple axel: \(3 \frac{1}{2}\) revolutions

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\sec \theta=-\frac{2 \sqrt{3}}{3}\) (b) \(\tan \theta=-\frac{\sqrt{3}}{3}\)

satellite in a circular orbit 1250 kilometers above Earth makes one complete revolution every 110 minutes. What is its linear speed? Assume that Earth is a sphere of radius 6378 kilometers.

Find two solutions of each equation. Give your solutions in both degrees \(\left(0^{\circ} \leq \theta < 360^{\circ}\right)\) and radians \((0 \leq \theta < 2 \pi) .\) Do not use a calculator. (a) \(\tan \theta=1\) (b) \(\sec \theta=\sqrt{2}\)

The circular blade on a saw has a diameter of 7.25 inches and rotates at 4800 revolutions per minute. (a) Find the angular speed of the blade in radians per minute. (b) Find the linear speed of the saw teeth (in inches per minute) as they contact the wood being cut.

A motorcycle wheel has a diameter of 19.5 inches (see figure) and rotates at 1050 revolutions per minute. (a) Find the angular speed in radians per minute. (b) Find the linear speed of the motorcycle (in inches per minute).

Prove the identity \(\arcsin (-x)=-\arcsin x\)

A Blu-ray disc is approximately 12 centimeters in diameter. The drive motor of the Blu-ray player is able to rotate up to 10,000 revolutions per minute, depending on what track is being read. (a) Find the maximum angular speed (in radians per second) of a Blu-ray disc as it rotates. (b) Find the maximum linear speed (in meters per second) of a point on the outermost track as the disc rotates.

Prove the identity \(\arctan (-x)=-\arctan x\)

The radii of the pedal sprocket, the wheel sprocket, and the wheel of the bicycle in the figure are 4 inches, 2 inches, and 14 inches, respectively. A cyclist is pedaling at a rate of 1 revolution per second. (a) Find the speed of the bicycle in feet per second and miles per hour. (b) Use your result from part (a) to write a function for the distance \(d\) (in miles) a cyclist travels in terms of the number \(n\) of revolutions of the pedal sprocket. (c) Write a function for the distance \(d\) (in miles) a cyclist travels in terms of time \(t\) (in seconds). Compare this function with the function from part (b).

Prove the identity arcsin \(x+\arccos x=\frac{\pi}{2}\)

Determine whether the statement is true or false. Justify your answer. A degree is a larger unit of measure than a radian.

Use the value of the trigonometric function to evaluate each function. \(\sin t=\frac{1}{3}\) (a) \(\sin (-t)\) (b) \(\csc (-t)\)

In calculus, it is shown that the area of the region bounded by the graphs of \(y=0, y=1 /\left(x^{2}+1\right), x=a,\) and \(x=b\) is given by Arca \(=\arctan b-\arctan a\) (see figure). Find the area for each value of \(a\) and \(b\) (a) \(a=0, b=1\) (b) \(a=-1, b=1\) (c) \(a=0, b=3\) (d) \(a=-1, b=3\)

Use the value of the trigonometric function to evaluate each function. \(\cos t=-\frac{3}{4}\) (a) \(\cos (-t)\) (b) \(\sec (-t)\)

Simplify the radical expression. \(\frac{4}{4 \sqrt{2}}\)

Determine whether the statement is true or false. Justify your answer. The angles of a triangle can have radian measures of \(2 \pi / 3, \pi / 4,\) and \(\pi / 12\).

Use the value of the trigonometric function to evaluate each function. \(\cos (-t)=-\frac{1}{5}\) (a) \(\cos t\) (b) \(\sec (-t)\)

Simplify the radical expression. \(\frac{2}{\sqrt{3}}\)

Use the value of the trigonometric function to evaluate each function. \(\sin (-t)=\frac{3}{8}\) (a) \(\sin t\) (b) \(\csc t\)

Prove that the area of a circular sector of radius \(r\) with central angle \(\theta\) is \(A=\frac{1}{2} r^{2} \theta,\) where \(\theta\) is measured in radians.

Simplify the radical expression. \(\frac{2 \sqrt{3}}{6}\)

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=30^{\circ}$$

Simplify the radical expression. \(\frac{5 \sqrt{5}}{2 \sqrt{10}}\)

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=60^{\circ}$$

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\sin \theta=\frac{5}{6}\)