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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 4 th quadrant, find sin ( t ) .

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If cos ( t ) = 2 9 and t is in the 1 st quadrant, find sin ( t ) .

77 9

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If sin ( t ) = 3 8 and t is in the 2 nd quadrant, find cos ( t ) .

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If sin ( t ) = 1 4 and t is in the 3 rd 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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Graph of circle with angle of t inscribed. Point of (0, 1) is at intersection of terminal side of angle and edge of circle.

sin t = 1 , cos t = 0

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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

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 is 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 last 6 minutes?

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Questions & Answers

For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Find the domain of the function in interval or inequality notation f(x)=4-9x+3x^2
prince Reply
hello
Jessica Reply
Outside temperatures over the course of a day can be modeled as a sinusoidal function. Suppose the high temperature of ?105°F??105°F? occurs at 5PM and the average temperature for the day is ?85°F.??85°F.? Find the temperature, to the nearest degree, at 9AM.
Karlee Reply
if you have the amplitude and the period and the phase shift ho would you know where to start and where to end?
Jean Reply
rotation by 80 of (x^2/9)-(y^2/16)=1
Garrett Reply
thanks the domain is good but a i would like to get some other examples of how to find the range of a function
bashiir Reply
what is the standard form if the focus is at (0,2) ?
Lorejean Reply
a²=4
Roy Reply
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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