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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

The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit 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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