<< Chapter < Page Chapter >> Page >

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
Got questions? Get instant answers now!
Got questions? Get instant answers now!
  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
Got questions? Get instant answers now!

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.

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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

( 1 2 , 3 2 )

Got questions? Get instant answers now!

Questions & Answers

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
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
Practice Key Terms 3

Get the best Algebra and trigonometry course in your pocket!





Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask