1 / 11

Chapter 7: Trigonometric Identities and Equations

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. .

jag
Download Presentation

Chapter 7: Trigonometric Identities and Equations

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  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 + 1 -1 = tan 2 X tan2 X + 1 = sec2 X • tan 2 X = tan 2 X Simplify

  6. Sum and Difference Identities • sin ( α + β)   =   sin αcosβ + cosα sin β • sin ( α − β)   =   sin αcosβ − cosα sin β • cos ( α + β)   =   cosαcosβ − sin α sin β • cos ( α − β)   =   cosαcosβ + 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 • cos2X=2cos²X-1 • cos2X=1-2sin²X • 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 cosX (sinX- ½)=0 Factor cos X = 0 or sinX- ½ =0 X= 90⁰ sinX= ½ X= 30⁰ Values are 30⁰ and 90⁰

More Related