Trigonometric Functions: The Unit Circle Section 4.2

1
Trigonometric Functions The Unit CircleSection
4.2
2
Objectives

• I can list the 6 trig functions
• I can find the key values of any of the trig
  functions on the Unit circle
• I can identify the period of each trig function
• I can identify which trig functions are even or
  odd

3
6 Trig Functions

• Cosine (cos)
• Sine (sin)
• Tangent (tan)
• Secant (sec)
• Cosecant (csc)
• Cotangent (cot)
• Which ones are related as reciprocals??

4
S O H - C A H - T O A
Parent functions
Reciprocal functions
5
Reciprocal Identities
6
Quotient Identities
7
We get cosine and sine values for angles from the
unit circle.We get the rest from SOH-CAH-TOA and
reciprocals
8
Evaluating Trig Functions
Use your unit circle, find the angle,
evaluate. Rationalize the denominator as needed.

• 1 Find the six trig. values for 300?.
• sin 300o csc 300o
• cos 300o sec 300o
• tan 300o cot 300o

9
Evaluating Trig Functions
Use your unit circle, find the angle,
evaluate. Rationalize the denominator as needed.

• 1 Find the six trig. values for -5p/4
• sin csc
• cos sec
• tan cot

10
Even and Odd Trigonometric Functions

• The cosine and secant functions are EVEN.
• cos(-t) cos t sec(-t) sec t
• The sine, cosecant, tangent, and cotangent
  functions are ODD.
• sin(-t) -sin t csc(-t) -csc t
• tan(-t) -tan t cot(-t) -cot t

11
Trig Properties
f(x) cos x
f(x) sin x
EVEN
ODD

12
sin(-t) -sin t
13
cos(-t) cos(t)
14
Problems
-1/4

• If sin (t) ¼, find sin (-t).
• If sin (t) is 3/8, find csc (-t).
• 3) If cos (t) -3/4, find cos(-t).

If sin (t) is 3/8, then csc (t) 8/3. We want to
find csc (-t) which is the opposite of csc (t)
-8/3.
cos(t) cos(-t) so -3/4
15
Definition of a Periodic Function

• A function f is periodic if there exists a
  positive number p such that
• f(t p) f(t)
• For all t in the domain of f. The smallest number
  p for which f is periodic is called the period of
  f.

16
Function Period (Radians) Period (Degrees)
Cosine 2p 360
Sine 2p 360
Secant 2p 360
Cosecant 2p 360
Tangent p 180
Cotangent p 180
17
Homework

• WS 8-7