1. Basic Trigonometric Identities and Equations

2. Trigonometric Identities
Quotient Identities
Reciprocal Identities
Pythagorean Identities
sin2q + cos2q = 1
tan2q + 1 = sec2q
cot2q + 1 = csc2q
sin2q = 1 - cos2q
tan2q = sec2q - 1
cot2q = csc2q - 1
cos2q = 1 - sin2q

3. Do you remember the Unit Circle?
Where did our pythagorean identities come from??
Do you remember the Unit Circle?
What is the equation for the unit circle?
x2 + y2 = 1
What does x = ? What does y = ?
(in terms of trig functions)
sin2θ + cos2θ = 1
Pythagorean Identity!

4. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by cos2θ
sin2θ + cos2θ =
cos2θ cos2θ cos2θ
tan2θ = sec2θ
Quotient
Identity
Reciprocal
Identity
another Pythagorean Identity

5. Take the Pythagorean Identity and discover a new one!
Hint: Try dividing everything by sin2θ
sin2θ + cos2θ =
sin2θ sin2θ sin2θ
cot2θ = csc2θ
Quotient
Identity
Reciprocal
Identity
a third Pythagorean Identity

6. Using the identities you now know, find the trig value.
1.) If cosθ = 3/4, find secθ 2.) If cosθ = 3/5, find cscθ.

7. 3.) sinθ = -1/3, find tanθ

8. Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify. b) a)

9. Simplifing Trigonometric Expressions
c) (1 + tan x)2 - 2 sin x sec x d)

10. Simplify each expression.

11. Simplifying trig Identity
Example: simplify tanxcosx
sin x
cos x
tanx cosx
tanxcosx = sin x

12. Simplifying trig Identity
sec x
csc x
Example: simplify 1
cos x 1
cos x
sinx = x
sec x
csc x 1
sin x =
sin x
cos x
= tan x

13. Example Simplify: = cot x (csc2 x - 1) Factor out cot x
= cot x (cot2 x)
Use pythagorean identity
= cot3 x
Simplify

14. Example Simplify: = sin x (sin x) + cos x Use quotient identity cos x
Simplify fraction with LCD
= sin2 x + (cos x)
cos x
= sin2 x + cos2x
cos x
Simplify numerator =
cos x
Use pythagorean identity
= sec x
Use reciprocal identity

15. *** Combine fraction Simplify the numerator Use pythagorean identity
Use Reciprocal Identity

16. Examples
Prove tan(x) cos(x) = sin(x)

17. Examples
Prove tan2(x) = sin2(x) cos-2(x)

18. Examples
Prove

19. Examples
Prove

20. Get common denominators
Hints for Establishing Identities
Get common denominators
If you have squared functions look for Pythagorean Identities
Work on the more complex side first
If you have a denominator of 1 + trig function try multiplying top & bottom by conjugate and use Pythagorean Identity
When all else fails write everything in terms of sines and cosines using reciprocal and quotient identities
It's like a puzzle, you can use identities and algebra to get them to match!