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Numeric

For the following exercises, state the reference angle for the given angle.

For the following exercises, find the reference angle, the quadrant of the terminal side, and the sine and cosine of each angle. If the angle is not one of the angles on the unit circle, use a calculator and round to three decimal places.

300°

60° , Quadrant IV, sin ( 300° ) = 3 2 , cos ( 300° ) = 1 2

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135°

45° , Quadrant II, sin ( 135° ) = 2 2 , cos ( 135° ) = 2 2

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120°

60° ,   Quadrant II, sin ( 120° ) = 3 2 , cos ( 120° ) = 1 2

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150°

30° , Quadrant II, sin ( 150° ) = 1 2 , cos ( 150° ) = 3 2

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7 π 6

π 6 , Quadrant III, sin ( 7 π 6 ) = 1 2 , cos ( 7 π 6 ) = 3 2

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3 π 4

π 4 , Quadrant II, sin ( 3 π 4 ) = 2 2 , cos ( 4 π 3 ) = 2 2

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2 π 3

π 3 , Quadrant II, sin ( 2 π 3 ) = 3 2 , cos ( 2 π 3 ) = 1 2

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7 π 4

π 4 , Quadrant IV, sin ( 7 π 4 ) = 2 2 , cos ( 7 π 4 ) = 2 2

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For the following exercises, find the requested value.

If cos ( t ) = 1 7 and t is in the fourth quadrant, find sin ( t ) .

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If cos ( t ) = 2 9 and t is in the first quadrant, find sin ( t ) .

77 9

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If sin ( t ) = 3 8 and t is in the second quadrant, find cos ( t ) .

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If sin ( t ) = 1 4 and t is in the third quadrant, find cos ( t ) .

15 4

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Find the coordinates of the point on a circle with radius 15 corresponding to an angle of 220° .

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Find the coordinates of the point on a circle with radius 20 corresponding to an angle of 120° .

( −10 ,   10 3 )

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Find the coordinates of the point on a circle with radius 8 corresponding to an angle of 7 π 4 .

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Find the coordinates of the point on a circle with radius 16 corresponding to an angle of 5 π 9 .

( –2.778 ,   15.757 )

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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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Graphical

For the following exercises, use the given point on the unit circle to find the value of the sine and cosine of t .

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

sin t = 1 2 , cos t = 3 2

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Graph of circle with angle of t inscribed. Point of (negative square root of 2 over 2, negative 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

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Graph of circle with angle of t inscribed. Point of (-1/2, 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

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Graph of circle with angle of t inscribed. Point of (square root of 2 over 2, negative 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

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Graph of circle with angle of t inscribed. Point of (-1,0) is at intersection of terminal side of angle and edge of circle.

sin t = 0 ,   cos t = 1

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Graph of circle with angle of t inscribed. Point of (0.803,-0.596 is at intersection of terminal side of angle and edge of circle.

sin t = 0.596 ,   cos t = 0.803

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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.

sin t = 1 2 , cos t = 3 2

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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.

sin t = 1 2 , cos t = 3 2

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Graph of circle with angle of t inscribed. Point of (-0.649, 0.761) is at intersection of terminal side of angle and edge of circle.

sin t = 0.761 , cos t = 0.649

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Technology

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

cos 5 π 9

−0.1736

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cos 3 π 4

−0.7071

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Extensions

For the following exercises, evaluate.

sin ( 11 π 3 ) cos ( 5 π 6 )

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sin ( 3 π 4 ) cos ( 5 π 3 )

2 4

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sin ( 4 π 3 ) cos ( π 2 )

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sin ( 9 π 4 ) cos ( π 6 )

6 4

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sin ( π 6 ) cos ( π 3 )

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sin ( 7 π 4 ) cos ( 2 π 3 )

2 4

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cos ( 5 π 6 ) cos ( 2 π 3 )

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cos ( π 3 ) cos ( π 4 )

2 4

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sin ( 5 π 4 ) sin ( 11 π 6 )

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sin ( π ) sin ( π 6 )

0

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Real-world applications

For the following exercises, use this scenario: A child enters a carousel that takes one minute to revolve once around. The child enters at the point ( 0 , 1 ) , that is, on the due north position. Assume the carousel revolves counter clockwise.

What are the coordinates of the child after 45 seconds?

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What are the coordinates of the child after 90 seconds?

( 0 , –1 )

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What are the coordinates of the child after 125 seconds?

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When will the child have coordinates ( 0.707 , –0.707 ) if the ride lasts 6 minutes? (There are multiple answers.)

37.5 seconds, 97.5 seconds, 157.5 seconds, 217.5 seconds, 277.5 seconds, 337.5 seconds

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When will the child have coordinates ( –0.866 , –0.5 ) if the ride lasts 6 minutes?

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Questions & Answers

can you solve it step b step
Ching Reply
what is linear equation with one unknown 2x+5=3
Joan Reply
-4
Joel
x=-4
Joel
x=-1
Joan
I was wrong. I didn't move all constants to the right of the equation.
Joel
x=-1
Cristian
Adityasuman x= - 1
Aditya
what is the VA Ha D R X int Y int of f(x) =x²+4x+4/x+2 f(x) =x³-1/x-1
Shadow Reply
can I get help with this?
Wayne
Are they two separate problems or are the two functions a system?
Wilson
Also, is the first x squared in "x+4x+4"
Wilson
x^2+4x+4?
Wilson
thank you
Wilson
Please see ***imgur.com/a/lpTpDZk for solutions
Wilson
f(x)=x square-root 2 +2x+1 how to solve this value
Marjun Reply
factor or use quadratic formula
Wilson
what is algebra
Ige Reply
The product of two is 32. Find a function that represents the sum of their squares.
Paul
if theta =30degree so COS2 theta = 1- 10 square theta upon 1 + tan squared theta
Martin Reply
how to compute this 1. g(1-x) 2. f(x-2) 3. g (-x-/5) 4. f (x)- g (x)
Yanah Reply
hi
John
hi
Grace
what sup friend
John
not much For functions, there are two conditions for a function to be the inverse function:   1--- g(f(x)) = x for all x in the domain of f     2---f(g(x)) = x for all x in the domain of g Notice in both cases you will get back to the  element that you started with, namely, x.
Grace
sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
acha se dhek ke bata sin theta ke value
Ajay
sin theta ke ja gha sin square theta hoga
Ajay
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
False statement so you cannot prove it
Wilson
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
Practice Key Terms 3

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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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