### Chapter 7: Trigonometric Identities and Equations

```Jami Wang
Period 3
Extra Credit PPT
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.
Pythagorean Identities cont.
 Example:
 1. If csc X = 4/3, find tan X
 csc2 X= 1 + cot2 X
 (4/3) 2 = 1 + cot2
 16/9 = 1 + cot2 X
 7/9 = cot2 X
 ±√7 / 3 = cot X
Find tan X
tan X= 1/ cot X
= ± (3 √7)/7
Pythagorean identity
Use 4/3 for csc X\
Verifying Trigonometric
Identities
1. Change to sin X / cos X
2. LCD
3. Factor (and Cancel)
4. Look Trig identities
5. Multiply by conjugate
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
 tan 2 X + 1 -1 = tan 2 X
X
 tan 2 X = tan 2 X
Multiply
tan2 X + 1 = sec2
Simplify
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)
Sum and Difference Identities cont.
 Tan 285⁰ = tan (240 ⁰ + 45 ⁰)
240 ⁰ and 45 ⁰are common
angles whose sum is 285⁰
= tan240 ⁰ + tan 45 ⁰ Sum Identity for Tangent
1-tan240 ⁰ tan45 ⁰
= √3+1
Multiply by conjugate to
simplify
1-(√3)(1)
= -2-√3
Double Angle Formulas
 sin2X= 2sinXcosX
 cos2X=cos²X-sin²X
 cos2X=2cos²X-1
 cos2X=1-2sin²X
 tan2X=2tanX
1-tan²X
Double Angle Formulas cont.
Example:
cos2X = cos²X-sin²X
= (√5/3)²-(2/3) ²
= 1/9
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
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
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