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 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

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 Multiply tan 2 X + 1 -1 = tan 2 X tan2 X + 1 = sec2 X tan 2 X = tan 2 X 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. 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

Double Angle Formulas sin2X= 2sinXcosX cos2X=cos²X-sin²X 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