Pre Cal Notes 5-3 Trigonometric Function on the Unit Circle
Unit Circle – a circle of radius 1 unit.
Consider the circle that I just gave you is a unit circle with center at the origin.
It’s equation is 𝑥 2 + 𝑦 2 = 1 .
1.
Use what you know to answer parts A and B.
You have already filled in the angle measures in radian and degrees. To determine the ordered pairs on the circle we need to use the idea that a point on the circle (x,y) is the point of intersection of the terminal side of the angle with the unit circle.
2.
Use the picture at the right to determine each of the six trig values in terms of x and y. a.
sin 𝜃 = b.
cos 𝜃 = c.
tan 𝜃 = d.
cot 𝜃 = e.
sec 𝜃 = f.
csc 𝜃 =
3.
Using the ratios that you just figured out above, answer part C on the unit circle sheet.
Since sin 𝜃 = 𝑦 𝑎𝑛𝑑 cos 𝜃 = 𝑥 are defined for every point on the unit circle and since every angle has exactly one pair of intersection points with the circle sin 𝜃 = 𝑦 𝑎𝑛𝑑 cos 𝜃 = 𝑥 are functions. The domain of both is All Real Numbers because they are defined for any angle. The range for both is [-1,1] because the y values are on a circle with r=1. Since you can’t divide by zero, the other four trig function have some angle measure excluded from their domains.
4.
Use the unit circle to find: a.
cos(−180) b.
sec 90 c.
sin −90 d.
cot 270
5.
Write the following example in the space below. Use the unit circle to find the value of all 6 trig functions for a 135 degree angle.

6.
Find the values of all 6 trig functions for a 210 degree angle.
You can also find the values of the six trig functions for an angle in standard position when the radius of the circle is not one.
8.
Write down the following example below: Find the values of the six trig function for angle 𝜃 in standard position when a point with coordinate (-15, 20) lies on the terminal side.
9.
What do you notice about the signs in the example above?
10.
Suppose 𝜃 is an angle in standard position whose terminal side lies in quadrant 4. If sec 𝜃 =
5
3 find the values of the remaining trig functions.
7.
Use your unit circle to fill the chart that I gave you for the six trig values at every angle.