1. Chapter 7: Trigonometric Identities and Equations
Jami Wang
Period 3
Extra Credit PPT

2. Pythagorean Identities
sin2 X + cos2 X = 1
tan2 X + 1 = sec2 X
1 + cot2 X = csc2 X
These identities can be used to help find values of trigonometric functions.

3. Pythagorean Identities cont.
Example:
1. If csc X = 4/3, find tan X
csc2 X= 1 + cot2 X Pythagorean identity
(4/3) 2 = 1 + cot2 Use 4/3 for csc X\
16/9 = 1 + cot2 X
7/9 = cot2 X
±√7 / 3 = cot X
Find tan X
tan X= 1/ cot X
= ± (3 √7)/7

4. Verifying Trigonometric Identities
1. Change to sin X / cos X
2. LCD
3. Factor (and Cancel)
4. Look Trig identities
5. Multiply by conjugate

5. Verifying Trigonometric Identities cont.
Example:
Verify that sec2 X – tan X cot X = tan 2 X is an identity
sec2 X – tan X * 1/tan X= tan 2 X cot X = 1/tan X
sec2 X – 1 = tan 2 X Multiply
tan 2 X = tan 2 X tan2 X + 1 = sec2 X
tan 2 X = tan 2 X Simplify

6. Sum and Difference Identities
sin ( α + β)   =   sin α cos β + cos α sin β
sin ( α − β)   =   sin α cos β − cos α sin β
cos ( α + β)   =   cos α cos β − sin α sin β
cos ( α − β)   =   cos α cos β + sin α sin β
tαn(α+β) = (tαnα + tαnβ)/(1 - tαnαtαnβ) tαn(α-β) = (tαnα - tαnβ)/(1 + tαnαtαnβ)

7. Sum and Difference Identities cont.
240 ⁰ and 45 ⁰are common angles whose sum is 285⁰
Sum Identity for Tangent
Multiply by conjugate to simplify
Tan 285⁰ = tan (240 ⁰ + 45 ⁰)
= tan240 ⁰ + tan 45 ⁰
1-tan240 ⁰ tan45 ⁰
= √3+1
1-(√3)(1)
= -2-√3

8. Double Angle Formulas sin2X= 2sinXcosX cos2X=cos²X-sin²X
tan2X=2tanX
1-tan²X

9. Double Angle Formulas cont.
Example:
cos2X = cos²X-sin²X
= (√5/3)²-(2/3) ²
= 1/9

10. Half Angle Formulas sin α /2 = ±√1-cos α/ 2 cos α/2 = ±√1+cos α/ 2
tan α/2 = ±√1-cos α/ 1+ cos α,
cos α≠-1

11. Solving Trigonometric Equations
Example:
sin X cos X – ½ cosX = 0
cos X (sinX- ½)=0 Factor
cos X = 0 or sinX- ½ =0
X= 90⁰ sinX= ½
X= 30⁰
Values are 30⁰ and 90⁰