1. Measuring Angles

In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 135°

In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 87.177°

In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 𝜋

In Exercises 1–6, the measure of an angle is given. Classify the angle as acute, right, obtuse, or straight. 𝜋/2

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

7𝜋 6

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

3𝜋 4

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

7𝜋 4

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

_ 2𝜋 3

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

_ 5𝜋 6

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

_ 5𝜋 4

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

_ 7𝜋 4

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

16𝜋 3

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

14𝜋 3

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

120°

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

-210°

In Exercises 41–56, use the circle shown in the rectangular coordinate system to draw each angle in standard position. State the quadrant in which the angle lies. When an angle's measure is given in radians, work the exercise without converting to degrees.

420°

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 395°

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -150°

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -760°

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 17𝜋 5

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. 25𝜋 6

In Exercises 57–70, find a positive angle less than or that is coterminal with the given angle. -𝜋 40

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.

A

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.

D

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.

F

Use the circle shown in the rectangular coordinate system to solve Exercises 81–86. Find two angles, in radians, between -2𝜋 and 2𝜋 such that each angle's terminal side passes through the origin and the given point.

E

In Exercises 87–90, find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time. 55 seconds

In Exercises 87–90, find the absolute value of the radian measure of the angle that the second hand of a clock moves through in the given time. 3 minutes and 40 seconds

In Exercises 8–12, draw each angle in standard position. 5𝜋 6

In Exercises 8–12, draw each angle in standard position. 8𝜋 3

In Exercises 8–12, draw each angle in standard position. -135°

In Exercises 13–17, find a positive angle less than 360° or 2𝜋 that is coterminal with the given angle. -445°

In Exercises 13–17, find a positive angle less than 360° or 2𝜋 that is coterminal with the given angle. 31𝜋 6

In Exercises 39–40, let θ be an angle in standard position. Name the quadrant in which θ lies. tan θ > 0 and sec θ > 0

In Exercises 39–40, let θ be an angle in standard position. Name the quadrant in which θ lies. tan θ > 0 and cos θ < 0

Find a positive angle less than 2𝜋 that is coterminal with 16𝜋 3

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places. sin 38°

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places. tan 32.7°

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places. csc 17°

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places. cos 𝜋 10

In Exercises 39–48, use a calculator to find the value of the trigonometric function to four decimal places. cot 𝜋 12

In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. sin θ = 0.2974

In Exercises 55–58, use a calculator to find the value of the acute angle θ to the nearest degree. tan θ = 4.6252

In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. cos θ = 0.4112

In Exercises 59–62, use a calculator to find the value of the acute angle θ in radians, rounded to three decimal places. tan θ = 0.4169

In Exercises 99–104, find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation. _ sin θ = √2 2

In Exercises 99–104, find two values of θ, 0 ≤ θ < 2𝜋, that satisfy each equation. _ sin θ = - √2 2

CONCEPT PREVIEW Fill in the blank to correctly complete each sentence. One degree, written 1°, represents ____________ of a complete rotation.

CONCEPT PREVIEW Fill in the blank to correctly complete each sentence. One minute, written 1' , is ________________ of a degree.

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 30°

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 45°

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 54°

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 10°

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 89°

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 14° 20'

Find the measure of (a) the complement and (b) the supplement of an angle with the given measure. See Examples 1 and 3. 50° 40' 50"

Find the measure of each marked angle. See Example 2.

Find the measure of each marked angle. See Example 2.

Find the measure of each marked angle. See Example 2.

Find the measure of each marked angle. See Example 2.

Find the measure of each marked angle. See Example 2.

Find the measure of each marked angle. See Example 2.

Find the measure of each marked angle. See Example 2 supplementary angles with measures 10𝓍 + 7 and 7𝓍 + 3 degrees

Find the measure of each marked angle. See Example 2 supplementary angles with measures 6𝓍 ― 4 and 8𝓍― 12 degrees

Find the measure of each marked angle. See Example 2 complementary angles with measures 9𝓍 + 6 and 3𝓍 degrees

Find the measure of each marked angle. See Example 2 complementary angles with measures 3𝓍 ― 5 and 6𝓍 ― 40 degrees

Find the measure of the smaller angle formed by the hands of a clock at the following times.

Find the measure of the smaller angle formed by the hands of a clock at the following times. 3:15

Find the measure of the smaller angle formed by the hands of a clock at the following times. 8:20

Perform each calculation. See Example 3. 75° 15' + 83° 32'

Perform each calculation. See Example 3. 47° 29' ― 71° 18'

Perform each calculation. See Example 3. 90° ― 51° 28'

Perform each calculation. See Example 3. 90° ― 17° 13'

Perform each calculation. See Example 3. 90° ― 36° 18' 47"

Perform each calculation. See Example 3. 55° 30' + 12° 44' ― 8° 15'

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example 4(a). 82° 30'

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example 4(a). 112° 15'

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example 4(a). ― 70° 48'

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example 4(a). 38° 42' 18"

Convert each angle measure to decimal degrees. If applicable, round to the nearest thousandth of a degree. See Example 4(a). 274° 18' 59"

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b). 174.255°

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b). ―25.485°

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b). 59.0854°

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b). 102.3771°

Convert each angle measure to degrees, minutes, and seconds. If applicable, round to the nearest second. See Example 4(b). 122.6853°

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. 26° 30'

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. ―203° 20'

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. 541°

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. ―541°

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. 1000°

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. 8440°

Find the angle of least positive measure (not equal to the given measure) that is coterminal with each angle. See Example 5. ―5280°

Give two positive and two negative angles that are coterminal with the given quadrantal angle. 90°

Give two positive and two negative angles that are coterminal with the given quadrantal angle. 0°

Give two positive and two negative angles that are coterminal with the given quadrantal angle. 270°

Write an expression that generates all angles coterminal with each angle. Let n represent any integer. 135°

Write an expression that generates all angles coterminal with each angle. Let n represent any integer. ―90°