Chapter 7 Trigonometric Identities and Equations Jami Wang

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

Pythagorean Identities �sin 2 X + cos 2 X = 1 �tan 2 X + 1 = sec 2 X � 1 + cot 2 X = csc 2 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 �csc 2 X= 1 + cot 2 X �(4/3) 2 = 1 + cot 2 � 16/9 = 1 + cot 2 X � 7/9 = cot 2 X �±√ 7 / 3 = cot X Find 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 sec 2 X – tan X cot X = tan 2 X is an identity � sec 2 X – tan X * 1/tan X= tan 2 X cot X = 1/tan X �sec 2 X – 1 = tan 2 X Multiply �tan 2 X + 1 -1 = tan 2 X + 1 = sec 2 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. �Tan 285⁰ = tan (240 ⁰ + 45 ⁰) 240 ⁰ and 45 ⁰are common angles whose sum is 285⁰ = tan 240 ⁰ + tan 45 ⁰ Sum Identity for Tangent 1 -tan 240 ⁰ tan 45 ⁰ = √ 3+1 Multiply by conjugate to simplify 1 -(√ 3)(1) = -2 -√ 3

Double Angle Formulas �sin 2 X= 2 sin. Xcos. X �cos 2 X=cos²X-sin²X �cos 2 X=2 cos²X-1 �cos 2 X=1 -2 sin²X �tan 2 X=2 tan. X 1 -tan²X

Double Angle Formulas cont. � Example: � cos 2 X = 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 cos. X = 0 cos X (sin. X- ½)=0 cos X = 0 or sin. X- ½ =0 X= 90⁰ sin. X= ½ X= 30⁰ Values are 30⁰ and 90⁰ Factor