<< 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. We know that

    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 6 ) = 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!

Questions & Answers

if three forces F1.f2 .f3 act at a point on a Cartesian plane in the daigram .....so if the question says write down the x and y components ..... I really don't understand
Syamthanda Reply
hey , can you please explain oxidation reaction & redox ?
Boitumelo Reply
hey , can you please explain oxidation reaction and redox ?
Boitumelo
for grade 12 or grade 11?
Sibulele
the value of V1 and V2
Tumelo Reply
advantages of electrons in a circuit
Rethabile Reply
we're do you find electromagnetism past papers
Ntombifuthi
what a normal force
Tholulwazi Reply
it is the force or component of the force that the surface exert on an object incontact with it and which acts perpendicular to the surface
Sihle
what is physics?
Petrus Reply
what is the half reaction of Potassium and chlorine
Anna Reply
how to calculate coefficient of static friction
Lisa Reply
how to calculate static friction
Lisa
How to calculate a current
Tumelo
how to calculate the magnitude of horizontal component of the applied force
Mogano
How to calculate force
Monambi
a structure of a thermocouple used to measure inner temperature
Anna Reply
a fixed gas of a mass is held at standard pressure temperature of 15 degrees Celsius .Calculate the temperature of the gas in Celsius if the pressure is changed to 2×10 to the power 4
Amahle Reply
How is energy being used in bonding?
Raymond Reply
what is acceleration
Syamthanda Reply
a rate of change in velocity of an object whith respect to time
Khuthadzo
how can we find the moment of torque of a circular object
Kidist
Acceleration is a rate of change in velocity.
Justice
t =r×f
Khuthadzo
how to calculate tension by substitution
Precious Reply
hi
Shongi
hi
Leago
use fnet method. how many obects are being calculated ?
Khuthadzo
khuthadzo hii
Hulisani
how to calculate acceleration and tension force
Lungile Reply
you use Fnet equals ma , newtoms second law formula
Masego
please help me with vectors in two dimensions
Mulaudzi Reply
how to calculate normal force
Mulaudzi
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply
Practice Key Terms 4

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

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

Ask