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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