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Algebraic

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

tan π 6

sec π 6

2 3 3

csc π 6

cot π 6

3

tan π 4

sec π 4

2

csc π 4

cot π 4

1

tan π 3

sec π 3

2

csc π 3

cot π 3

3 3

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

tan 5 π 6

sec 7 π 6

2 3 3

csc 11 π 6

cot 13 π 6

3

tan 7 π 4

sec 3 π 4

2

csc 5 π 4

cot 11 π 4

−1

tan 8 π 3

sec 4 π 3

−2

csc 2 π 3

cot 5 π 3

3 3

tan 225°

sec 300°

2

csc 150°

cot 240°

3 3

tan 330°

sec 120°

−2

csc 210°

cot 315°

−1

If sin t = 3 4 , and t is in quadrant II, find cos t , sec t , csc t , tan t , cot t .

If cos t = 1 3 , and t is in quadrant III, find sin t , sec t , csc t , tan t , cot t .

If sin t = 2 2 3 , sec t = 3 , csc t = 3 2 4 , tan t = 2 2 , cot t = 2 4

If tan t = 12 5 , and 0 t < π 2 , find sin t , cos t , sec t , csc t , and cot t .

If sin t = 3 2 and cos t = 1 2 , find sec t , csc t , tan t , and cot t .

sec t = 2 , csc t = 2 3 3 , tan t = 3 , cot t = 3 3

If sin 40° 0.643 cos 40° 0.766 sec 40° , csc 40° , tan 40° , and cot 40° .

If sin t = 2 2 , what is the sin ( t ) ?

2 2

If cos t = 1 2 , what is the cos ( t ) ?

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

3.1

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

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

1.4

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

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

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

Technology

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

csc 5 π 9

cot 4 π 7

–0.228

sec π 10

tan 5 π 8

–2.414

sec 3 π 4

csc π 4

1.414

tan 98°

cot 33°

1.540

cot 140°

sec 310°

1.556

Extensions

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

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

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

sin ( t ) 0.79

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

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

csc t 1.16

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

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

even

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

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

even

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

csc t tan t

sec t csc t

sin t cos t = tan t

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.

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

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?

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

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° .

Practice Key Terms 6

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Source:  OpenStax, Essential precalculus, part 2. OpenStax CNX. Aug 20, 2015 Download for free at http://legacy.cnx.org/content/col11845/1.2
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