1. Pyrhagorean Identities
U5 Day 12.5

2. What is an Identity? Equations are conditional.
Ex. 7 = 2x Only true if…
BUT… an Identity is ALWAYS TRUE, regardless of the value of the variable.
Ex. 2(x-1) = 2x – True if x is…

3. Reciprocal Identities
sin x = csc x =
cos x = sec x =
tan x = cot x =
You already know these!

4. Quotient Identities
tan θ =
So, cot θ =
See how many you know already?

5. Pythagorean Identities
sin2x + cos2x = 1
Is this always true?
If x is any angle?
That’s because…
YES!
YES!

6. So… sin2x + cos2x = 1 In the unit circle, x2 + y2 = 1 y = sinq q 1
X = cos q
In the unit circle, x2 + y2 = 1
So… sin2x + cos2x = 1

7. Divide everything by sin2x
What about this?
Divide everything by sin2x
sin2x + cos2 x = 1
___________________ ________________ ______________
sin2x sin2x sin2x
cot2x = csc2x
Another identity…
This is the same as… sin2x + cos2x = 1

8. How about this one? Divide everything by cos2x sin2x + cos2 x = 1
________________ ________________ ______________
cos2x cos2x cos2x
tan2x = sec2x
Another identity…
This is the same as… sin2x + cos2x = 1

9. Pythagorean Identities
sin2x + cos2x = 1
1 + tan2x = sec2x
1 + cot2x = csc2x

10. Use identities to find sin x, cos x, tan x
Given sin x = -2/3 and x is in quadrant III, find cos x and tan x.
Or…make a triangle

11. Simplifying Expressions! Ex. 1
sin x cos x – sin x
sin x (cos x – 1)

12. Factoring Trig Expressions Ex.2

13. Factoring Trig Expressions Ex.3

14. Simplifying Expressions Ex.4
= csc θ
Hint: get a common denominator

15. Remember the Pythagorean Identity! sin2x + cos2x = 1