1. Section 6.2 The Unit Circle and Circular Functions
Chapter 6
Section 6.2
The Unit Circle and Circular Functions

2. 1 Trigonometric Functions
If t is a real number that is the length of an arc on the unit circle and P is a point at the end of that arc with coordinates (x,y) then we get the following expressions for each of the trigonometric ratios:
P(x,y) 1 t
Find the values of the trigonometric functions at t = /6. 1

3. Find the values of the trigonometric functions at t = 3/2.1
This concept for the trigonometric functions agrees with what we learned about the trigonometric ratios. This is a special case where the point on the terminal side of the angle in standard position is on the unit circle. Notice x is the length of the adjacent side and y is the length of the opposite side and the hypotenuse is of length 1. 1

4. The values of the trigonometric functions will be the same as that of the trigonometric ratios. In particular for the angles 0,/6,/4,/3,/2 (i.e. 0°,30°,45°,60°,90°). In the picture to the right the first quadrant is shown along with the terminal points on the unit circle. 1 - 2

5. Reference Arc
A reference arc is similar to a reference angle. The reference number for a given number is the shortest distance you would need to travel to get to the x-axis on either the positive or negative side. The reference numbers can be used to calculate the coordinates of various points on the unit circle. The reference number for t is sometimes denoted 𝑡 . x y P
sin 2𝜋 3 = x y P
sin 5𝜋 4 = − 2 2
cos 2𝜋 3 = −1 2
cos 5𝜋 4 = − 2 2
tan 2𝜋 3 =− 3
tan 5𝜋 4 =1
The reference arc for 2𝜋 3 is the number 𝜋 3
The point P has coordinates given by −1 2 ,
The reference arc for 5𝜋 4 is the number 𝜋 4
The point P has coordinates given by − , − 2 2

6. Reciprocal Identities:
Pythagorean Identities:
tan 𝑡 = sin 𝑡 cos 𝑡 𝑎𝑛𝑑 cot 𝑡 = cos 𝑡 sin 𝑡
Sine and Cosine Identities:
Even & Odd Properties
Changing the direction of the angle from clockwise to counterclockwise or visa versa makes no difference to the cosine and secant, they are called even functions. The sine, tangent, cotangent and cosecant a change in direction makes them the negative of what they were.