1. Section 4.2
The Unit Circle

2. About the unit circle
A unit circle is a circle with a radius of 1 whose center is at the origin.
The standard equation of every unit circle is

3. A Picture

4. The Wrapping Function
In a unit circle, the measure of a central angle is equal to the length of the intercepted arc.
It is important to note that this measure is a real number.
Let P(s) = (x,y) be a function that has, as its input, an arc length s measured from (1,0) and, as its output, the ordered pair on the unit circle at the other endpoint of s.

5. Six Trigonometric Functions
Sine: sin s = y
Cosine: cos s = x
Tangent: tan s = y/x
Cosecant: csc s = 1/y
Secant: sec s = 1/x
Cotangent: cot s = x/y

6. A Picture

7. Examples—Find the six trig functions

8. Trig Function Properties: Even and Odd
Cosine and Secant are Even:
cos(-t) = cos t
sec(-t) = sec t
Sine, Cosecant, Tangent, and Cotangent are Odd:
sin(-t) = -sin t csc(-t) = -csc t
tan(-t) = -tan t cot(-t) = -cot t

9. Reciprocal Identities

10. Quotient Identities

11. Pythagorean Identities

12. Periodicity
Periodicity refers to the tendency of a function to repeat itself.
Sine and Cosine have period 2”pi”.
Tangent and Cotangent have period “pi”.

13. Using Your Calculator
Know how to convert your scientific calculator from DEGREE mode to RADIAN mode.
Use your calculator’s reciprocal function to get values for SEC, CSC, and COT.