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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

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit 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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