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A 23-ft ladder leans against a building so that the angle between the ground and the ladder is 80° . How high does the ladder reach up the side of the building?

22.6506 ft

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The angle of elevation to the top of a building in New York is found to be 9 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

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The angle of elevation to the top of a building in Seattle is found to be 2 degrees from the ground at a distance of 2 miles from the base of the building. Using this information, find the height of the building.

368.7633 ft

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Assuming that a 370-foot tall giant redwood grows vertically, if I walk a certain distance from the tree and measure the angle of elevation to the top of the tree to be 60° , how far from the base of the tree am I?

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Review exercises

Angles

For the following exercises, convert the angle measures to degrees.

For the following exercises, convert the angle measures to radians.

Find the length of an arc in a circle of radius 7 meters subtended by the central angle of 85°.

10.385 meters

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Find the area of the sector of a circle with diameter 32 feet and an angle of 3 π 5 radians.

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For the following exercises, find the angle between 0° and 360° that is coterminal with the given angle.

For the following exercises, find the angle between 0 and 2 π in radians that is coterminal with the given angle.

20 π 11

2 π 11

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For the following exercises, draw the angle provided in standard position on the Cartesian plane.

Find the linear speed of a point on the equator of the earth if the earth has a radius of 3,960 miles and the earth rotates on its axis every 24 hours. Express answer in miles per hour.

1036.73 miles per hour

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A car wheel with a diameter of 18 inches spins at the rate of 10 revolutions per second. What is the car's speed in miles per hour?

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Unit Circle: Sine and Cosine Functions

Find the exact value of sin π 3 .

3 2

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Find the exact value of cos π 4 .

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Find the exact value of cos π .

–1

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State the reference angle for 300° .

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State the reference angle for 3 π 4 .

π 4

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Compute cosine of 330° .

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Compute sine of 5 π 4 .

2 2

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State the domain of the sine and cosine functions.

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State the range of the sine and cosine functions.

[ 1 , 1 ]

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The Other Trigonometric Functions

For the following exercises, find the exact value of the given expression.

For the following exercises, use reference angles to evaluate the given expression.

If sec ( t ) = 2.5 , what is the sec ( t ) ?

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If tan ( t ) = 0.6 , what is the tan ( t ) ?

0.6

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If tan ( t ) = 1 3 , find tan ( t π ) .

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If cos ( t ) = 2 2 , find sin ( t + 2 π ) .

2 2 or 2 2

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Which trigonometric functions are even?

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Which trigonometric functions are odd?

sine, cosecant, tangent, cotangent

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Right Triangle Trigonometry

For the following exercises, use side lengths to evaluate.

cos ( π 2 ) = sin ( __° )

0

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csc ( 18 ° ) = sec ( __° )

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For the following exercises, use the given information to find the lengths of the other two sides of the right triangle.

cos B = 3 5 , a = 6

b = 8 , c = 10

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For the following exercises, use [link] to evaluate each trigonometric function.

A right triangle with side lengths of 11 and 6. Corners A and B are also labeled.

For the following exercises, solve for the unknown sides of the given triangle.

A 15-ft ladder leans against a building so that the angle between the ground and the ladder is 70° . How high does the ladder reach up the side of the building?

14.0954 ft

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The angle of elevation to the top of a building in Baltimore is found to be 4 degrees from the ground at a distance of 1 mile from the base of the building. Using this information, find the height of the building.

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Practice test

Convert 5 π 6 radians to degrees.

150°

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Convert −620° to radians.

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Find the length of a circular arc with a radius 12 centimeters subtended by the central angle of 30° .

6.283 centimeters

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Find the area of the sector with radius of 8 feet and an angle of 5 π 4 radians.

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Find the angle between and 360° that is coterminal with 375° .

15°

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Find the angle between 0 and 2 π in radians that is coterminal with 4 π 7 .

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Draw the angle 315° in standard position on the Cartesian plane.

A graph of a circle with an angle inscribed.
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Draw the angle π 6 in standard position on the Cartesian plane.

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A carnival has a Ferris wheel with a diameter of 80 feet. The time for the Ferris wheel to make one revolution is 75 seconds. What is the linear speed in feet per second of a point on the Ferris wheel? What is the angular speed in radians per second?

3.351 feet per second, 2 π 75 radians per second

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Find the exact value of sin π 6 .

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Compute sine of 240° .

3 2

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State the domain of the sine and cosine functions.

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State the range of the sine and cosine functions.

[ 1 , 1 ]

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Find the exact value of cot π 4 .

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Find the exact value of tan π 3 .

3

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Use reference angles to evaluate csc 7 π 4 .

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Use reference angles to evaluate tan 210 ° .

3 3

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If csc t = 0.68 , what is the csc ( t ) ?

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If cos t = 3 2 , find cos ( t 2 π ) .

3 2

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Which trigonometric functions are even?

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Find the missing angle: cos ( π 6 ) = sin ( ___ )

π 3

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Find the missing sides of the triangle A B C : sin B = 3 4 , c = 12

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Find the missing sides of the triangle.

A right triangle with hyptenuse length of 9 and angle measure of 60 degrees.

a = 9 2 , b = 9 3 2

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The angle of elevation to the top of a building in Chicago is found to be 9 degrees from the ground at a distance of 2000 feet from the base of the building. Using this information, find the height of the building.

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Practice Key Terms 5

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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