1. Unit 3: Trigonometric Identities
MA3A5. Students will establish the identities and use them to simplify trigonometric expressions and verify equivalence statements.
LG 3-1 Simplifying & Verifying Identities
LG 3-2 Applying Trig Identities
Test WEDNESDAY 9/26
You will be using the internet to find the identities for this unit. We will talk about verifying these identities tomorrow. You will need to memorize these, so start NOW  Keep these identities in your notes. Print them out on one sheet of paper (front only) using the “handout” option.

2. Introduction - Complete the Doublets (word identities) and the Trigonometry Triangles (back)
Answer Key:
#1 Change TEN to TWO #2 Change TEN to SIX.
T E N T E N
T o N T i N
T O o S I x
T W o S I X
#3 Change FIVE to FOUR #4 Change PIG to STY
F I V E P I G
F I l E w I G
F i l L w A G
f a l l w A Y
f a i l s A Y
F O I L S T Y
F O u L
F O U R

3. Answer Key continued…
#5 Change TEN to ONE. T E N t i n a metal t i p tilt l i p edge l i d cover a i d help a n d with (a conjuntion) a n t insect a c t perform a c e a card a r e a form of "to be" o r e mineral O N E

4. Answer Key continued… Reciprocal Identities Product Identities
Pythagorean Identities

5. What is a trigonometric identity?
A trigonometric identity is a trigonometric equation that is valid for all values of the variables for which the expression is defined.
In this unit, you will be manipulating expressions to make them equal something
When simplifying, you won’t know the answer
When verifying, you have the answer and your job is to manipulate one side of an equation to make it look like the other side

6. Unit 3: Trig Identities
You will now make a booklet with all the identities you are required to learn for this unit.
PLEASE PAY ATTENTION as we fold our booklets
On the FRONT COVER, write the title of the unit and YOUR NAME!
Do NOT lose your booklet!
Look for this symbol so you know when to write in your booklet
Booklet

7. Reciprocal Identities
Booklet pg 1
Reciprocal Identities
Also work with powers…

8. Booklet pg 2
Quotient Identities

9. Pythagorean Identities
Draw a right triangle on a separate sheet of paper. Label one of the angles as . Label the hypotenuse with a length of 1 unit. Label the side opposite  as a and the side adjacent to  as b.
Use the right-triangle ratio for sine to write an equation for sin(). Solve this equation for a.
Use the right-triangle ratio for cosine to write an equation for cos(). Solve this equation for b.
Write the formula for Pythagorean Theorem. Then substitute your expressions for a and b above and substitute the length of your hypotenuse for c into the Pythagorean Formula. What you now have is the Pythagorean Identity for Sine and Cosine.

10. Pythagorean Identities
cos2 + sin2 = 1
+ 1
tan2
= sec2
+ 1
cot2
= csc2
Each of the three Pythagorean Identities can be rearranged into two additional forms by simply adding or subtracting terms on both sides of the equation. Do this now with all three identities. You should have a total of 9 formulas in your booklet.

11. Pythagorean Identities
Booklet pgs 3-4
Pythagorean Identities
cos2 + sin2 = 1
+ 1
tan2
= sec2
+ 1
cot2
= csc2
Each of the three Pythagorean Identities can be rearranged into two additional forms by simply adding or subtracting terms on both sides of the equation. Do this now with all three identities >. You should have a total of 9 formulas.

12. Sum and Difference Identities
Booklet pg 5
Sum and Difference Identities
The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like:
The identity above is a short hand method for writing two identities as one. When these identities are broken up, they look like:

13. Double-Angle Identities
Booklet pg 6
Double-Angle Identities
sin (2x) = 2sin x cos x
cos (2x) = cos2x - sin2x
= 2cos2x – 1
= 1 - 2sin2x

14. Problem Solving Strategies
Booklet (back cover)
Problem Solving Strategies
Create a monomial denominator
Add fractions (find an LCD)
Factor if possible
Convert everything to sines and cosines
When verifying, work on ONE side only – always pick the more complicated side to transform
Always try SOMETHING! It may not be the right thing but it’s better than nothing!

15. Example 1: Simplify sin x cot x
Substitute using the quotient property.
Simplify.
Done!

16. Example 2:
Simplify
Use the reciprocal and quotient properties to make fractions.
Cancel the sines and cosines.
Done!

17. Example 3:
Simplify:

18. Simplify

19. Simplify

20. Simplify