1. Basic Trigonometric Identities and Equations
Pre-Calculus
Teacher – Mrs.Volynskaya

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
5.4.3

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 4.) secθ = -7/5, find sinθ

8. Simplifying Trigonometric Expressions
Identities can be used to simplify trigonometric expressions.
Simplify. b) a)
5.4.5

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

10. Simplify each expression.

11. Simplifying trig Identity
Example1: simplify tanxcosx
sin x
cos x
tanx cosx
tanxcosx = sin x

12. Simplifying trig Identity
sec x
csc x
Example2: simplify 1
cos x 1
cos x
sinx = x
sec x
csc x 1
sin x =
sin x
cos x
= tan x

13. Simplifying trig Identity
cos2x - sin2x
cos x
Example2: simplify
= sec x
cos2x - sin2x
cos x
cos2x - sin2x 1

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

15. 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

16. Your Turn! Combine fraction Simplify the numerator
Use pythagorean identity
Use Reciprocal Identity

17. Practice 1
cos2θ
cosθ
sin2θ cos2θ
secθ-cosθ
csc2θ
cotθ
tan2θ

18. One way to use identities is to simplify expressions involving trigonometric functions. Often a good strategy for doing this is to write all trig functions in terms of sines and cosines and then simplify. Let’s see an example of this:
substitute using each identity
simplify

19. Another way to use identities is to write one function in terms of another function. Let’s see an example of this:
This expression involves both sine and cosine. The Fundamental Identity makes a connection between sine and cosine so we can use that and solve for cosine squared and substitute.

20. (E) Examples
Prove tan(x) cos(x) = sin(x)

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

22. (E) Examples
Prove

23. (E) Examples
Prove