5.1. The Unit Circle http://www.ck12.org
5.1 The Unit Circle
Here you will use your knowledge of basic triangle trigonometry to identify key points and angles around a circle of
radius one centered at the origin.
Theunit circleis a circle of radius one, centered at the origin, that summarizes all the 30-60-90 and 45-45-90 triangle
relationships that exist. When memorized, it is extremely useful for evaluating expressions like cos( 135 ◦)or
sin(−^53 π). It also helps to produce the parent graphs of sine and cosine.
How can you use the unit circle to evaluate cos( 135 ◦)and sin(−^53 π)?
Watch This
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/61156http://www.youtube.com/watch?v=i56P6xzsB5Y James Sousa: Determine Trigonometric Function Values Using
the Unit Circle
Guidance
You already know how to translate between degrees and radians and the triangle ratios for 30-60-90 and 45-45-
90 right triangles. In order to be ready to completely fill in and memorize a unit circle, two triangles need to be
worked out. Start by finding the side lengths of a 30-60-90 triangle and a 45-45-90 triangle each with hypotenuse
equal to 1.
TABLE5.1:
30 ◦ 60 ◦ 90 ◦
x x√
3 2 x(^12)
√ 3
2 1
TABLE5.2:
45 ◦ 45 ◦ 90 ◦
x x x√
√^2
22 √ 22 1
This is enough information to fill out the important points in the first quadrant of the unit circle. The values of the
xandycoordinates for each of the key points are shown below. Remember that thexandycoordinates come from
the lengths of the legs of the special right triangles, as shown specifically for the 30◦angle. Always remember to