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Algebraic

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

tanπ6

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secπ6

233

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cscπ6

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cotπ6

3

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tanπ4

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secπ4

2

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cscπ4

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cotπ4

1

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tanπ3

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secπ3

2

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cscπ3

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cotπ3

33

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

tan5π6

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sec7π6

233

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csc11π6

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cot13π6

3

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tan7π4

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sec3π4

2

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csc5π4

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cot11π4

−1

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tan8π3

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sec4π3

−2

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csc2π3

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cot5π3

33

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tan225°

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sec300°

2

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csc150°

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cot240°

33

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tan330°

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sec120°

−2

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csc210°

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cot315°

−1

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If sint=34, and t is in quadrant II, find cost,sect,csct,tant,cott.

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If cost=13, and t is in quadrant III, find sint,sect,csct,tant,cott.

If sint=223,sect=3,csct=324,tant=22,cott=24

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If tant=125, and 0t<π2, find sint,cost,sect,csct, and cott.

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If sint=32 and cost=12, find sect,csct,tant, and cott.

sect=2,csct=233,tant=3,cott=33

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If sin40°0.643cos40°0.766sec40°,csc40°,tan40°,andcot40°.

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If sint=22, what is the sin(t)?

22

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If cost=12, what is the cos(t)?

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If sect=3.1, what is the sec(t)?

3.1

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If csct=0.34, what is the csc(t)?

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If tan t = 1.4 , what is the tan ( t ) ?

1.4

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If cot t = 9.23 , what is the cot ( t ) ?

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Graphical

For the following exercises, use the angle in the unit circle to find the value of the each of the six trigonometric functions.

Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 2 2 , cos t = 2 2 , tan t = 1 , cot t = 1 , sec t = 2 , csc t = 2

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Graph of circle with angle of t inscribed. Point of (-1/2, negative square root of 3 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 3 2 , cos t = 1 2 , tan t = 3 , cot t = 3 3 , sec t = 2 , csc t = 2 3 3

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Technology

For the following exercises, use a graphing calculator to evaluate.

csc 5 π 9

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cot 4 π 7

–0.228

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sec π 10

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tan 5 π 8

–2.414

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sec 3 π 4

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csc π 4

1.414

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tan 98°

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cot 33°

1.540

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cot 140°

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sec 310°

1.556

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Extensions

For the following exercises, use identities to evaluate the expression.

If tan ( t ) 2.7 , and sin ( t ) 0.94 , find cos ( t ) .

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If tan ( t ) 1.3 , and cos ( t ) 0.61 , find sin ( t ) .

sin ( t ) 0.79

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If csc ( t ) 3.2 , and cos ( t ) 0.95 , find tan ( t ) .

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If cot ( t ) 0.58 , and cos ( t ) 0.5 , find csc ( t ) .

csc t 1.16

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Determine whether the function f ( x ) = 2 sin x cos x is even, odd, or neither.

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Determine whether the function f ( x ) = 3 sin 2 x cos x + sec x is even, odd, or neither.

even

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Determine whether the function f ( x ) = sin x 2 cos 2 x is even, odd, or neither.

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Determine whether the function f ( x ) = csc 2 x + sec x is even, odd, or neither.

even

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For the following exercises, use identities to simplify the expression.

csc t tan t

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sec t csc t

sin t cos t = tan t

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Real-world applications

The amount of sunlight in a certain city can be modeled by the function h = 15 cos ( 1 600 d ) , where h represents the hours of sunlight, and d is the day of the year. Use the equation to find how many hours of sunlight there are on February 10, the 42 nd day of the year. State the period of the function.

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The amount of sunlight in a certain city can be modeled by the function h = 16 cos ( 1 500 d ) , where h represents the hours of sunlight, and d is the day of the year. Use the equation to find how many hours of sunlight there are on September 24, the 267 th day of the year. State the period of the function.

13.77 hours, period: 1000 π

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The equation P = 20 sin ( 2 π t ) + 100 models the blood pressure, P , where t represents time in seconds. (a) Find the blood pressure after 15 seconds. (b) What are the maximum and minimum blood pressures?

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The height of a piston, h , in inches, can be modeled by the equation y = 2 cos x + 6 , where x represents the crank angle. Find the height of the piston when the crank angle is 55° .

7.73 inches

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The height of a piston, h , in inches, can be modeled by the equation y = 2 cos x + 5 , where x represents the crank angle. Find the height of the piston when the crank angle is 55° .

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Practice Key Terms 6

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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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