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Using reference angles to find sine and cosine

  1. Using a reference angle, find the exact value of cos ( 150° ) and sin ( 150° ) .
  2. Using the reference angle, find cos 5 π 4 and sin 5 π 4 .
  1. 150° is located in the second quadrant. The angle it makes with the x -axis is 180° 150° = 30° , so the reference angle is 30° .

    This tells us that 150° has the same sine and cosine values as 30° , except for the sign.

    cos ( 30° ) = 3 2 and sin ( 30° ) = 1 2

    Since 150° is in the second quadrant, the x -coordinate of the point on the circle is negative, so the cosine value is negative. The y -coordinate is positive, so the sine value is positive.

    cos ( 150° ) = 3 2 and sin ( 150° ) = 1 2
  2. 5 π 4 is in the third quadrant. Its reference angle is 5 π 4 π = π 4 . The cosine and sine of π 4 are both 2 2 . In the third quadrant, both x and y are negative, so:
    cos 5 π 4 = 2 2 and sin 5 π 4 = 2 2
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  1. Use the reference angle of 315° to find cos ( 315° ) and sin ( 315° ) .
  2. Use the reference angle of π 6 to find cos ( π 6 ) and sin ( π 6 ) .
  1. cos ( 315° ) = 2 2 ,   sin ( 315° ) = 2 2
  2. cos ( π 6 ) = 3 2 , sin ( π 6 ) = 1 2
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Using reference angles to find coordinates

Now that we have learned how to find the cosine and sine values for special angles in the first quadrant, we can use symmetry and reference angles to fill in cosine and sine values for the rest of the special angles on the unit circle    . They are shown in [link] . Take time to learn the ( x , y ) coordinates of all of the major angles in the first quadrant.

Graph of unit circle with angles in degrees, angles in radians, and points along the circle inscribed.
Special angles and coordinates of corresponding points on the unit circle

In addition to learning the values for special angles, we can use reference angles to find ( x , y ) coordinates of any point on the unit circle, using what we know of reference angles along with the identities

x = cos  t y = sin  t

First we find the reference angle corresponding to the given angle. Then we take the sine and cosine values of the reference angle, and give them the signs corresponding to the y - and x -values of the quadrant.

Given the angle of a point on a circle and the radius of the circle, find the ( x , y ) coordinates of the point.

  1. Find the reference angle by measuring the smallest angle to the x -axis.
  2. Find the cosine and sine of the reference angle.
  3. Determine the appropriate signs for x and y in the given quadrant.

Using the unit circle to find coordinates

Find the coordinates of the point on the unit circle at an angle of 7 π 6 .

We know that the angle 7 π 6 is in the third quadrant.

First, let’s find the reference angle by measuring the angle to the x -axis. To find the reference angle of an angle whose terminal side is in quadrant III, we find the difference of the angle and π .

7 π 6 π = π 6

Next, we will find the cosine and sine of the reference angle.

cos ( π 6 ) = 3 2 sin ( π 6 ) = 1 2

We must determine the appropriate signs for x and y in the given quadrant. Because our original angle is in the third quadrant, where both x and y are negative, both cosine and sine are negative.

cos ( 7 π 6 ) = 3 2 sin ( 7 π ) = 1 2

Now we can calculate the ( x , y ) coordinates using the identities x = cos θ and y = sin θ .

The coordinates of the point are ( 3 2 , 1 2 ) on the unit circle.

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Find the coordinates of the point on the unit circle at an angle of 5 π 3 .

( 1 2 , 3 2 )

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Questions & Answers

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.
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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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