Chapter 7: Trigonometric Identities and Equations

Jami WangPeriod 3

Extra Credit PPT

Pythagorean Identitiessin2 X + cos2 X = 1tan2 X + 1 = sec2 X1 + 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 Pythagorean identity

(4/3) 2 = 1 + cot2 Use 4/3 for csc X\16/9 = 1 + cot2 X7/9 = cot2 X±√7 / 3 = cot XFind tan Xtan X= 1/ cot X

= ± (3 √7)/7

Verifying Trigonometric Identities1. Change to sin X / cos X2. LCD3. Factor (and Cancel)4. Look Trig identities5. 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 Xsec2 X – 1 = tan 2 X Multiplytan 2 X + 1 -1 = tan 2 X tan2 X

+ 1 = sec2 Xtan 2 X = tan 2 X Simplify

Sum and Difference Identitiessin ( α + β)   =   sin α cos β + cos α sin β sin ( α − β)   =   sin α cos β − cos α sin βcos ( 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) = tan240 ⁰ + tan 45 ⁰

1-tan240 ⁰ tan45 ⁰

= √3+1 1-(√3)(1)= -2-√3

240 ⁰ and 45 ⁰are common angles whose sum is 285⁰

Sum Identity for Tangent

Multiply by conjugate to simplify

Double Angle Formulassin2X= 2sinXcosXcos2X=cos²X-sin²Xcos2X=2cos²X-1cos2X=1-2sin²Xtan2X=2tanX 1-tan²X

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

Half Angle Formulassin α /2 = ±√1-cos α/ 2cos α/2 = ±√1+cos α/ 2tan α/2 = ±√1-cos α/ 1+ cos α,

cos α≠-1

Solving Trigonometric EquationsExample: sin X cos X – ½ cosX = 0

cos X (sinX- ½)=0 Factorcos X = 0 or sinX- ½ =0X= 90⁰ sinX= ½

X= 30⁰

Values are 30⁰ and 90⁰