1. Section 3.6 Basic Trigonometric Identities
© Copyright all rights reserved to Homework depot:

2. I) What is a Trigonometric Identity?
A trigonometric identity is an equation that is equal for all values of the variable(s) for which the equation is defined
Examples of trigonometric identities
Trigonometric equations that are not Identities
Both sides are equal for all values of “x”
Graphically, if both sides overlap each other completely, then the equation is an identity
Equation is true only at certain values of “x”
Graphically, both sides only intersect at certain points, then the equation is NOT an identity
© Copyright all rights reserved to Homework depot:

3. II) Odd vs Even Identities:
Even Identities: An function that looks the same when reflected over the y-axis (Horizontal Reflection)
Odd Identities: A function that looks the same when reflected over both the X and Y axis x y
-2p -p p 2p -1 1 x y
-2p -p p 2p -1 1
© Copyright all rights reserved to Homework depot:

4. III) Pythagorean Identities:
Review: The coordinates of any point on the circumference of an unit circle can be represented by:
X-coordinate
Y-coordinate
Since:
Pythagorean Identity
Other Pythagorean Identities can be generated by dividing all terms by either “cos2x” or “sin2x”
© Copyright all rights reserved to Homework depot:

5. IV) Basic Identities Odd- Even Identities Reciprocal Identities
Quotient Identities
Pythagorean Identities
Other identities can be created by manipulating the equation
© Copyright all rights reserved to Homework depot:

6. V) Verifying and Proving Identities
There are two ways to Verify an identity
Plug variety of numbers into the equation
If the equation is equal for all the values, then the equation is an identity OR
Graph the equations, if they completely overlap, then it’s an identity
Proving an Identity
Simplify the equation algebraically and then show that both sides are equal
When proving algebraically, first convert all functions into sine or cosine
Then simplify using basic identities
© Copyright all rights reserved to Homework depot:

7. Ex: Verify the Following identity:
Since the equation is equal for both verifications, then it’s likely to be an identity
NUMERICALLY:
Pick a random number
Pick another number
Both sides are equal!
Both sides are equal again!
GRAPHICALLY: x y -4 -3 -2 -1 1 2 3 4
Since the graphs overlap each other
completely, then it
must be an identity
© Copyright all rights reserved to Homework depot:

8. Practice: Verify the following identities Numerically:
Pick a random number for “x”
Make “x” = 10 rad.
Both sides are equal
Both sides are equal
Try another value for “x”
Make “x” = 2 rad.
Since the equation is equal for both verifications, then it is likely to be an identity
The equation is likely to
be an identity
© Copyright all rights reserved to Homework depot:

9. Practice: Verify each of the following Identities:
Numerically:
Numerically:
Not an Identity!!
Graphically:
Graphically: x y
-2p -p p 2p -2 -1 1 2 x y
-2p -p p 2p -2 -1 1 2
© Copyright all rights reserved to Homework depot:

10. vi) Proving Identities Algebraically:
When proving trigonometric identities:
Convert all trig. functions to sine or cosine
Use basic trig. Identities to simplify complicated ones
Odd/Even, Quotient, Pythagorean Identities
Start with the side that looks “more complicated”
You may need to rationalize the expression, factor out common factors, or multiply all terms by the LCD
Trial and Error (Do whatever it takes)
Once the left side and right side are equal then the equation is a trigonometric identity
© Copyright all rights reserved to Homework depot:

11. Vii) Proving Trigonometric Identities By Using Basic Identities:
Since the left side looks more complicated, we will prove the identity from the left
Cancel out common factors
Quotient Identity!
© Copyright all rights reserved to Homework depot:

12. viii) Proving by Adding/Subtracting Identities
You can also prove the identity from the other side
Split into 2 separate fractions
Common Denominator
Combine the
Fractions
The proof may be the different
but the result is the same
© Copyright all rights reserved to Homework depot:

13. Practice: Prove the following identities:
Get the LCD with the numerator
Common Denominator
Factor out sin θ
Combine the
Fractions
© Copyright all rights reserved to Homework depot:

14. ix) Proving Identities Using Fractions :
Pythagorean
Identities
Divide the fractions
© Copyright all rights reserved to Homework depot:

15. x) Proving Identities by Factoring:
Difference of squares
Factor out the sine function
Pythagorean identity
Common Denominator
Pythagorean Identity
Expand
FOIL
© Copyright all rights reserved to Homework depot:

16. Practice: Prove the following Identities by Factoring
Bracket the last 2 terms
Pythagorean Identity
Factor the Trinomial
Pythagorean identity
Factor: Difference
of squares
© Copyright all rights reserved to Homework depot:

17. xi) Prove by Conjugating The Expression:
Quotient Identities
Reciprocal Identity
Multiply both top & bottom by the conjugate
Expand
Pythagorean Identity
Pythagorean identity
Formula Sheet
© Copyright all rights reserved to Homework depot:

18. Practice: Prove the following Identity by conjugating the expression:
Formula Sheet
Quotient Identities
Reciprocal Identity
Multiply both top & bottom by the conjugate
Expand
Pythagorean Identity
Pythagorean identity
CLOSE
© Copyright all rights reserved to Homework depot:

19. Homework:
Assignment 3.6