Section 1
Angles and Their Measure
Explain the difference between a positive angle and a negative angle.
Explain the difference between complementary angles and supplementary angles.
Would it be better to measure angles by dividing the circumference of a circle into 100 equal parts rather than 360 equal parts as in degree measure? Explain.
Explain the connection between an angle of 1 radian and the radius of a circle.
Find the degree measure to two decimal places for each angle in Problems $51-56 .$$$3.07$$
You are watching your nieces ride a Ferris wheel. Explain how you could do a mental calculation to estimate their angular speed.
Refer to Problem 5. Explain how you could do a mental calculation to estimate their linear speed.
Find the degree measure of each of the angles in Problems $7-12$, keeping in mind that an angle of one complete rotation corresponds to $360^{\circ}$$$\frac{1}{4} \text { rotation }$$
Find the degree measure of each of the angles in Problems $7-12$, keeping in mind that an angle of one complete rotation corresponds to $360^{\circ}$$$\frac{1}{5} \text { rotation }$$
Find the degree measure of each of the angles in Problems $7-12$, keeping in mind that an angle of one complete rotation corresponds to $360^{\circ}$$$\frac{3}{4} \text { rotation }$$
Find the degree measure of each of the angles in Problems $7-12$, keeping in mind that an angle of one complete rotation corresponds to $360^{\circ}$$$\frac{3}{8} \text { rotation }$$
Find the degree measure of each of the angles in Problems $7-12$, keeping in mind that an angle of one complete rotation corresponds to $360^{\circ}$$$\text { - } \frac{9}{\text { rotations }}$$
Find the degree measure of each of the angles in Problems $7-12$, keeping in mind that an angle of one complete rotation corresponds to $360^{\circ}$$$\frac{7}{6} \text { rotations }$$
Find the radian measure of a central angle $\theta$ opposite an are s in a circle of radius $r$, where $\mathrm{r}$ and $\mathrm{s}$ are as given in Problems $13-16 .$$$r=4 \text { centimeters, } s=24 \text { centimeters }$$
Find the radian measure of a central angle $\theta$ opposite an are s in a circle of radius $r$, where $\mathrm{r}$ and $\mathrm{s}$ are as given in Problems $13-16 .$$$r=8 \text { inches, } s=16 \text { inches }$$
Find the radian measure of a central angle $\theta$ opposite an are s in a circle of radius $r$, where $\mathrm{r}$ and $\mathrm{s}$ are as given in Problems $13-16 .$$$r=12 \text { feet, } s=30 \text { feet }$$
Find the radian measure of a central angle $\theta$ opposite an are s in a circle of radius $r$, where $\mathrm{r}$ and $\mathrm{s}$ are as given in Problems $13-16 .$$$r=18 \text { meters, } s=27 \text { meters }$$
Find the radian measure of each angle in Problems $17-22,$ keeping in mind that an angle of one complete rotation corresponds to $2 \pi$ radians.$$\frac{1}{8} \text { rotation }$$
Find the radian measure of each angle in Problems $17-22,$ keeping in mind that an angle of one complete rotation corresponds to $2 \pi$ radians.$$\frac{1}{6} \text { rotation }$$
Find the radian measure of each angle in Problems $17-22,$ keeping in mind that an angle of one complete rotation corresponds to $2 \pi$ radians.$$\frac{3}{4} \text { rotation }$$
Find the radian measure of each angle in Problems $17-22,$ keeping in mind that an angle of one complete rotation corresponds to $2 \pi$ radians.$$5 \cdot \frac{5}{12} \text { rotation }$$
Find the radian measure of each angle in Problems $17-22,$ keeping in mind that an angle of one complete rotation corresponds to $2 \pi$ radians.$$\frac{5}{12} \text { rotation }$$
Find the radian measure of each angle in Problems $17-22,$ keeping in mind that an angle of one complete rotation corresponds to $2 \pi$ radians.$$\frac{11}{8} \text { rotations }$$
Find the exact radian measure, in terms of $\pi,$ of each angle in Problems $23-26$.$$30^{\circ}, 60^{\circ}, 90^{\circ}, 120^{\circ}, 150^{\circ}, 180^{\circ}$$
Find the exact radian measure, in terms of $\pi,$ of each angle in Problems $23-26$.$$60^{\circ}, 120^{\circ}, 180^{\circ}, 240^{\circ}, 300^{\circ}, 360^{\circ}$$
Find the exact radian measure, in terms of $\pi,$ of each angle in Problems $23-26$.$$-45^{\circ},-90^{\circ},-135^{\circ},-180^{\circ}$$
Find the exact radian measure, in terms of $\pi,$ of each angle in Problems $23-26$.$$-90^{\circ},-180^{\circ},-270^{\circ},-360^{\circ}$$
Find the exact degree measure of each angle in Problems $27-30 .$$$\frac{\pi}{3}, \frac{2 \pi}{3}, \pi, \frac{4 \pi}{3}, \frac{5 \pi}{3}, 2 \pi$$
Find the exact degree measure of each angle in Problems $27-30 .$$$\frac{\pi}{6}, \frac{\pi}{3}, \frac{\pi}{2}, \frac{2 \pi}{3}, \frac{5 \pi}{6}, \pi$$
Find the exact degree measure of each angle in Problems $27-30 .$$$-\frac{\pi}{2},-\pi,-\frac{3 \pi}{2},-2 \pi$$
Find the exact degree measure of each angle in Problems $27-30 .$$$-\frac{\pi}{4},-\frac{\pi}{2},-\frac{3 \pi}{4},-\pi$$
In Problems $31-36$, determine whether the statement is true or false. If true, explain why. If false, give a counterexample.If two angles in standard position have the same measure, then they are coterminal.
In Problems $31-36$, determine whether the statement is true or false. If true, explain why. If false, give a counterexample.If two angles in standard position are coterminal, then they have the same measure.
In Problems $31-36$, determine whether the statement is true or false. If true, explain why. If false, give a counterexample.If two positive angles are complementary, then both are acute.
In Problems $31-36$, determine whether the statement is true or false. If true, explain why. If false, give a counterexample.\text { If two positive angles are complementary, then both are acute. }
In Problems $31-36$, determine whether the statement is true or false. If true, explain why. If false, give a counterexample.If the terminal side of an angle in standard position lies in quadrant I, then the angle is positive.
In Problems $31-36$, determine whether the statement is true or false. If true, explain why. If false, give a counterexample.If the initial and terminal sides of an angle coincide, then the measure of the angle is zero.
Convert each angle in Problems $37-40$ to decimal degrees to three decimal places.$$5^{\circ} 51^{\prime} 33^{\prime \prime}$$
Convert each angle in Problems $37-40$ to decimal degrees to three decimal places.$$14^{\circ} 18^{\prime} 37^{\prime \prime}$$
Convert each angle in Problems $37-40$ to decimal degrees to three decimal places.$$354^{\circ} 8^{\prime} 29^{n}$$
Convert each angle in Problems $37-40$ to decimal degrees to three decimal places.$$184^{\circ} 31^{\prime} 7^{\prime \prime}$$
Convert each angle in Problems $41-44$ to degree-minute-second form.$$3.042^{\circ}$$
Convert each angle in Problems $41-44$ to degree-minute-second form.$$49.715^{\circ}$$
Convert each angle in Problems $41-44$ to degree-minute-second form.$$403.223^{\circ}$$
Convert each angle in Problems $41-44$ to degree-minute-second form.$$156.808^{\circ}$$
Find the radian measure to three decimal places for each angle in Problems $45-50 .$$$64^{\circ}$$
Find the radian measure to three decimal places for each angle in Problems $45-50 .$$$25^{\circ}$$
Find the radian measure to three decimal places for each angle in Problems $45-50 .$$$108.413^{\circ}$$
Find the radian measure to three decimal places for each angle in Problems $45-50 .$$$203.097^{\circ}$$
Find the radian measure to three decimal places for each angle in Problems $45-50 .$$$13^{\circ} 25^{\prime} 14^{\prime \prime}$$
Find the radian measure to three decimal places for each angle in Problems $45-50 .$$$56^{\circ} 11^{\prime} 52^{\prime \prime}$$
Find the degree measure to two decimal places for each angle in Problems $51-56 .$$$0.93$$
Find the degree measure to two decimal places for each angle in Problems $51-56 .$$$0.08$$
Find the degree measure to two decimal places for each angle in Problems $51-56 .$$$1.13$$
Find the degree measure to two decimal places for each angle in Problems $51-56 .$$$-2.35$$
Find the degree measure to two decimal places for each angle in Problems $51-56 .$$$-1.72$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$250^{\circ}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$150^{\circ}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$275^{\circ}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$195^{\circ}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$\frac{3 \pi}{4}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$\frac{\pi}{3}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$\frac{3 \pi}{2}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$\frac{7 \pi}{4}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$-330^{\circ}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$-450^{\circ}$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$-1.5$$
Indicate whether each angle in Problems $57-68$ is a first-, second-, third $,$ or fourth-quadrant angle or a quadrantal angle. All angles are in standard position in a rectangular coordinate system. (A sketch may be of help in some problems.)$$-4$$
Verbally describe the meaning of a central angle in a circle with radian measure 1 .
Verbally describe the meaning of an angle with degree measure 1 .
In Problems $71-74,$ find all angles $\theta$ in degree measure that satisfy the given conditions.$$360^{\circ} \leq \theta \leq 720^{\circ} \text { and } \theta \text { is coterminal with } 210^{\circ}$$
In Problems $71-74,$ find all angles $\theta$ in degree measure that satisfy the given conditions.$$360^{\circ} \leq \theta \leq 720^{\circ} \text { and } \theta \text { is coterminal with } 120^{\circ}$$
In Problems $71-74,$ find all angles $\theta$ in degree measure that satisfy the given conditions.$$-360^{\circ} \leq \theta \leq 0^{\circ} \text { and } \theta \text { is coterminal with } 45^{\circ}$$
In Problems $71-74,$ find all angles $\theta$ in degree measure that satisfy the given conditions.$$-360^{\circ} \leq \theta \leq 0^{\circ} \text { and } \theta \text { is coterminal with } 280^{\circ}$$
In Problems $75-78$, find all angles $\theta$ in radian measure that satisfy the given conditions.$$2 \pi \leq \theta \leq 6 \pi \text { and } \theta \text { is coterminal with } \pi / 3$$
In Problems $75-78$, find all angles $\theta$ in radian measure that satisfy the given conditions.$$2 \pi \leq \theta \leq 6 \pi \text { and } \theta \text { is coterminal with } 5 \pi / 4$$
In Problems $75-78$, find all angles $\theta$ in radian measure that satisfy the given conditions.$$-4 \pi \leq \theta \leq 0 \text { and } \theta \text { is coterminal with } \pi$$
In Problems $75-78$, find all angles $\theta$ in radian measure that satisfy the given conditions.$$-4 \pi \leq \theta \leq 0 \text { and } \theta \text { is coterminal with } \pi / 6$$
CIRCUMFERENCE OF THE EARTH The early Greeks used the proportion $s / C=\theta^{\circ} / 360^{\circ},$ where $s$ is an arc length on a circle, $\theta^{\circ}$ is degree measure of the corresponding central angle, and $C$ is the
CIRCUMFERENCE OF THE EARTH Repeat Problem 79 with the sun crossing the vertical pole in Alexandria at $7^{\circ} 12^{\prime}$.
CIRCUMFERENCE OF THE EARTH In Problem 79 , verbally explain how $\theta$ in the figure was determined.
CIRCUMFERENCE OF THE EARTH Verbally explain how the radius, surface area, and volume of the Earth can be determined from the result of Problem $79 .$
ANGULAR SPEED A wheel with diameter 6 feet makes 200 revolutions per minute. Find the angular speed (in radians per second) and the linear speed (in feet per second) of a point on the rim.
ANGULAR SPEED A point on the rim of a wheel with diameter 6 feet has a linear speed of 100 feet per second. Find the angular speed (in radians per second) and the number of revolutions per minute.
RADIAN MEASURE What is the radian measure of the larger angle made by the hands of a clock at $4: 30 ?$ Express the answer exactly in terms of $\pi$.
RADIAN MEASURE What is the radian measure of the smaller angle made by the hands of a clock at $1: 30 ?$ Express the answer exactly in terms of $\pi$.
ENGINEERING Through how many radians does a pulley of 10-centimeter diameter turn when 10 meters of rope are pulled through it without slippage?
ENGINEERING Through how many radians does a pulley of 6 -inch diameter turn when 4 feet of rope are pulled through it without slippage?
ASTRONOMY A line from the sun to the Earth sweeps out an angle of how many radians in 1 week? Assume the Earth's orbit is circular and there are 52 weeks in a year. Express the answer in terms of $\pi$ and as a decimal to two decimal places.
ASTRONOMY A line from the center of the Earth to the equator sweeps out an angle of how many radians in 9 hours? Express the answer in terms of $\pi$ and as a decimal to two decimal places.
ENGINEERING A trail bike has a front wheel with a diameter of 40 centimeters and a back wheel of diameter 60 centimeters. Through what angle in radians does the front wheel turn if the back wheel turns through 8 radians?
ENGINEERING In Problem 91 , through what angle in radians will the back wheel turn if the front wheel turns through 15 radians?
ANGULAR SPEED If the trail bike of Problem 91 travels at a speed of 10 kilometers per hour, find the angular speed (in radians per second) of each wheel.
ANGULAR SPEED If a car travels at a speed of 60 miles per hour, find the angular speed (in radians per second) of a tire that has a diameter of 2 feet.
ASTRONOMY The sun is about $9.3 \times 10^{7}$ mi from the Earth. If the angle subtended by the diameter of the sun on the surface of the Earth is $9.3 \times 10^{-3} \mathrm{rad}$, approximately what is the diameter of the sun to the nearest thousand miles in standard decimal notation?
. ASTRONOMY The moon is about 381,000 kilometers from the Earth. If the angle subtended by the diameter of the moon on the surface of the Earth is 0.0092 radians, approximately what is the diameter of the moon to the nearest hundred kilometers?
PHOTOGRAPHY The angle of view of a 1,000 -millimeter telephoto lens is $2.5^{\circ}$. At 750 feet, what is the width of the field of view to the nearest foot?
PHOTOGRAPHY The angle of view of a 300 -millimeter lens is $8^{\circ}$. At 500 feet, what is the width of the field of view to the nearest foot?