<< 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

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
HERVE Reply
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
Oliver Reply
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
Kwesi Reply
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
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.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit 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