The Unit Circle
This is VERY important!!!
We will be working with this through next nine weeks
What is the Unit Circle
• The unit circle gives us a
graphical way to represent
all trig functions
• sin, cos, tan, csc, sec, cot
• It is called the unit circle
because the radius of the
circle is 1 (or 1 unit)
• This is a completed unit
circle, we are going to
break it down into pieces
to learn.
How the Unit Circle Works
(x,y)
1
y
x
• Each part of the unit circle
you saw on the last slide is
built from this image. If we
inscribe a right triangle in a
circle with radius of 1 we
can describe our trig
functions in terms of x and
y.
• Find the following values
using the graph:
• sin(θ) =
• cos(θ) =
• tan(θ) =
Angles in the Unit Circle
• We use the following angles in the first quadrant of our unit circle:
• 300 , 450 , and 600
• We use these angles because they come from the special right
triangles:
30-60-90 Triangle
45-45-90 Triangle
Modified Special Right Triangles
• Since the radius of a unit circle is 1 we have to change the values on
our special right triangles to reflect a hypotenuse of 1.
45-45-90 Triangle
30-60-90 Triangle
45-45-90 for Unit
Circle
30-60-90 for
Unit Circle
1
1
The Quadrants
• The unit circle is VERY repetitive! Each of the four quadrants has the
same x and y values, with different signs. What you need to
remember is the sign (+ or -) for each x and y in each quadrant:
Lets Build the
Unit Circle
We will start with the first
quadrant, because the other 3
use the same values.
Each angle needs to be labeled
with the following information:
• X – value of end point
• Remember this is our
cosine value
• Y – value of end point
• Remember this is our
sine value
• Angle measure in degrees
• Angle measure in Radians
Now
Quadrant 2
Pay Attention to the numbers
in the first quadrant!
Each angle needs to be labeled
with the following information:
• X – value of end point
• Remember this is our
cosine value
• Y – value of end point
• Remember this is our
sine value
• Angle measure in degrees
• Angle measure in Radians
Now
Quadrant 3
Pay Attention to the numbers
in the first quadrant!
Each angle needs to be labeled
with the following information:
• X – value of end point
• Remember this is our
cosine value
• Y – value of end point
• Remember this is our
sine value
• Angle measure in degrees
• Angle measure in Radians
Now
Quadrant 4
Pay Attention to the numbers
in the first quadrant!
Each angle needs to be labeled
with the following information:
• X – value of end point
• Remember this is our
cosine value
• Y – value of end point
• Remember this is our
sine value
• Angle measure in degrees
• Angle measure in Radians
Rules for Basic Trig Function on the Unit Circle
Let θ be the angle at the origin. Then:
sin(θ) = y
cos(θ) = x
𝑦
tan θ =
𝑥
Please write this on your unit circle!
Assignment
From Earlier this Week:
Textbook Page 252 Problems 1-11 odd
Textbook Page 306 Problems 3-29 odd
RED BOOK Page 366 Problems 71, 75, 79, and 83
From Today:
Trig Worksheet 1