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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

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
which of these functions is not uniformly continuous on 0,1
Basant Reply
solve this equation by completing the square 3x-4x-7=0
Jamiz Reply
X=7
Muustapha
=7
mantu
x=7
mantu
3x-4x-7=0 -x=7 x=-7
Kr
x=-7
mantu
9x-16x-49=0 -7x=49 -x=7 x=7
mantu
what's the formula
Modress
-x=7
Modress
new member
siame
what is trigonometry
Jean Reply
deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
Thomas
solve for me this equational y=2-x
Rubben Reply
what are you solving for
Alex
solve x
Rubben
you would move everything to the other side leaving x by itself. subtract 2 and divide -1.
Nikki
then I got x=-2
Rubben
it will b -y+2=x
Alex
goodness. I'm sorry. I will let Alex take the wheel.
Nikki
ouky thanks braa
Rubben
I think he drive me safe
Rubben
how to get 8 trigonometric function of tanA=0.5, given SinA=5/13? Can you help me?m
Pab Reply
More example of algebra and trigo
Stephen Reply
What is Indices
Yashim Reply
If one side only of a triangle is given is it possible to solve for the unkown two sides?
Felix Reply
cool
Rubben
kya
Khushnama
please I need help in maths
Dayo Reply
Okey tell me, what's your problem is?
Navin
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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