1. Section 8.4: Trig Identities & Equations
Pre-Calculus

2. 8.4 Trig Identities & Equations
Objectives:
Identify the relationship of trig functions and positive and negative angles
Identify the Pythagorean trig relationships
Identify the cofunction trig relationships
Apply various trig relationships to simplify expressions.
Vocabulary: sine, cosine, tangent, cosecant, secant, cotangent, cofunction

3. Review of Reciprocal Trig Relationships
𝑡𝑎𝑛θ= sin θ cos θ

4. Example 1: Simplifying Expressions
Simplify the following Expressions

5. Part 1: Pythagorean Trig Relationships
Let’s take a look at the unit circle.
Using the Pythagorean Theorem, how can you relate all three sides of the triangle?
sin2θ + cos2θ = 1
This is one of the Pythagorean
Trig Relationships

6. Examples: Simplifying Expressions

7. Part 1: Pythagorean Trig Relationships
Starting with sin2θ + cos2θ = 1, how can you manipulate it to get other following Pythagorean Trig Relationships?
1 + tan2θ = sec2θ
Divide both sides by cos2θ
1 + cot2θ = csc2θ
Divide both sides by sin2θ
These are the final 2 of the 3 Pythagorean Trig Relationships

8. Examples: Simplifying Expressions

9. Part 2: Cofunction Trig Relationships
Sine & Cosine, Tangent & Cotangent, Secant & Cosecant are all Cofunctions.
Trig Cofunctions have the following relationship
WHY?

10. Examples: Simplifying Expressions
Simplify the following
tan (90° – A) =
Cos (π/2 – x) =

11. Part 3: Trig Relationships with Negative & Positive Angles
Let’s take a look at a positive and negative angle on the unit circle

12. Part 3: Trig Relationships with Negative & Positive Angles
Let’s take a look at sin θ. What does this equal according to our picture?
What about sin –θ. What does this equal according to our picture?
What can we say about the relationship between sin θ & sin –θ?

13. Part 3: Trig Relationships With Negative and Positive Angles
We just proved that sin (-θ) = - sin θ
What do you think the relationship between cos (- θ) and cos θ is?
cos (- θ) = cos θ
What about the relationship between tan (- θ) and tan θ?
tan (- θ) = - tan θ

14. Part 3: Trig Relationships With Negative and Positive Angles
Let’s look at csc (- θ) and csc θ. What is the relationship?
csc (- θ) = - csc θ
What about the relationship between sec (- θ) and sec θ?
sec (- θ) = sec θ
What about the relationship between cot (- θ) and cot θ?
cot (- θ) = - cot θ

15. Examples: Practice Simplifying
Write the equivalent trig function with a positive angle
Sin (-π/2)
Cos (-π/3)
Cot (-3π/4)

16. Suggestions Change everything on both sides to sine and cosine.
Start with the more complicated side
Try substituting basic identities (changing all functions to be in terms of sine and cosine may make things easier)
Try algebra: factor, multiply, add, simplify, split up fractions
If you’re really stuck make sure to:
Change everything on both sides to sine and cosine.
Work with only one side at a time!

17. Don’t Get Discouraged! Every identity is different
Keep trying different approaches
The more you practice, the easier it will be to figure out efficient techniques
If a solution eludes you at first, sleep on it! Try again the next day. Don’t give up!
You will succeed!

18. Tips to help simplify expressions
There are 4 different categories of trig relationships which each have different key components to look for
Reciprocal Relationships
Most commonly used in some type of format similar to
cot y · sin y
manipulating a fraction with trig functions
Usually the functions aren’t squared when they are in this format
Negative/Positive Angle Relationships
Similar to the example problems previously in this powerpoint
tan (-45°)

19. Tips to help simplify expressions
There are 4 different categories of trig relationships which each have different key components to look for
Cofunction Relationships
Similar to the example problems previously in this powerpoint
cos (90° – A)
Pythagorean Relationships (MOST COMMON/CHALLENGING!)
Includes exponents to the second degree
Includes expanding two binomials
Addition and subtraction of fractions
May need to factor out a trig function before simplifying
Or some type of variation of the previous

20. Tips to help simplify expressions
Though most of the problems are separated into their respective categories, you may find yourself having to combine multiple relationships to fully simplify an expression.
Maybe you’ll start with Pythagorean relationships, then to fully simplify you may use Reciprocal relationships.
In most cases, fully simplifying an expression will leave the expression with only one term

21. Homework
Textbook pg 321: #1, 5, 13, 21, 31