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

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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

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

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

2 2

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

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

3.1

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If csc t = 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.

This is an image of a 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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This is an image of a 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 to three decimal places.

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.

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 267th 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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Chapter 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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Questions & Answers

the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
e power cos hyperbolic (x+iy)
Vinay Reply
10y
Michael
tan hyperbolic inverse (x+iy)=alpha +i bita
Payal Reply
prove that cos(π/6-a)*cos(π/3+b)-sin(π/6-a)*sin(π/3+b)=sin(a-b)
Tejas Reply
why {2kπ} union {kπ}={kπ}?
Huy Reply
why is {2kπ} union {kπ}={kπ}? when k belong to integer
Huy
if 9 sin theta + 40 cos theta = 41,prove that:41 cos theta = 41
Trilochan Reply
what is complex numbers
Ayushi Reply
Please you teach
Dua
Yes
ahmed
Thank you
Dua
give me treganamentry question
Anshuman Reply
Solve 2cos x + 3sin x = 0.5
shobana Reply
Practice Key Terms 6

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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