3.06 - Trig Identities

University:
High School
Course:
High School Physics
Academic year:

2024
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Trigonometry 3.06: Trigonometric Identities Identity O A rule that is true for all values of the domain. O A comparison of expressions with no numeric solution. O Goal: simplify to show that both sides are equal to each other. Basic Identities The ones we’ve already seen Trigonometric Ratios 𝑜𝑝𝑝 sin 𝜃 = ℎ𝑦𝑝 𝑎𝑑𝑗 cos 𝜃 = ℎ𝑦𝑝 𝑜𝑝𝑝 sin 𝜃 tan 𝜃 = = 𝑎𝑑𝑗 cos 𝜃 Reciprocal Ratios 1 sec 𝜃 = cos 𝜃 1 csc 𝜃 = sin 𝜃 1 cos 𝜃 cot 𝜃 = = tan 𝜃 sin 𝜃 The Development of THREE New Identities So you don’t have to prove them everytime Identity One 𝑃(cos 𝜃 , sin 𝜃) 1 Pythagorean Theorem 𝑐 2 = 𝑎2 + 𝑏 2 12 = cos 𝜃 sin 𝜃 2 + sin 𝜃 **NOTE** sin 𝜃 2 = sin2 𝜃 cos 𝜃 sin 𝜃 Id1 2 ≠ sin 𝜃 2 1 = cos2 𝜃 + sin2 𝜃 2 Identity Two With identities, always work on one side to make it look like the other O Divide both sides of ID1 by cos2(ϴ) O cos2 𝜃 + sin2 𝜃 = 1 cos2 𝜃 sin2 𝜃 1 + = 2 2 cos 𝜃 cos 𝜃 cos 2 𝜃 cos 2 𝜃 sin 𝜃 2 1 2 +( ) =( ) 2 cos 𝜃 cos 𝜃 cos 𝜃 Id2 1 + tan2 𝜃 = sec 2 𝜃 Identity Three O Divide ID1 by sin2(ϴ). sin2 𝜃 + cos 2 𝜃 = 1 sin2 𝜃 cos 2 𝜃 1 + = 2 2 sin 𝜃 sin 𝜃 sin2 𝜃 sin2 𝜃 sin2 𝜃 Id3 +( cos 𝜃 2 1 2 ) =( ) sin 𝜃 sin 𝜃 1 + cot 2 𝜃 = csc 2 𝜃 Pythagorean Identities 2 2 1. cos 𝜃 + sin 𝜃 = 1 2 2 2 2 2. 1 + tan 𝜃 = sec 𝜃 3. 1 + cot 𝜃 = csc 𝜃 How to Complete Identities O REDUCE the most complicated side by simplifying as much as possible O Analyze connections until left side matches right. Things to look for … The analyzing process could include: O Basic Identities (4 definitions) O 3 Squared Identities: only for 2nd degree! O Factoring O Number Sense O Operations Examples 1 1 + = 1 sec 2 𝜃 csc 2 𝜃 O By Laws of Exponents 2 1 sec 𝜃 + 2 1 =1 csc 𝜃 O By Reciprocal Ratios cos 𝜃 2 + sin 𝜃 2 =1 O By ID 1 1=1 2 2 2 2 tan 𝜃 − sin 𝜃 = sin 𝜃 tan 𝜃 O By tan2 𝜃 sin2 𝜃 cos2 𝜃 = sin2 𝜃 2𝜃 − sin cos 2 𝜃 O Common Denominator sin2 𝜃 sin2 𝜃 cos 2 𝜃 − 2 cos 𝜃 cos 2 𝜃 O Bring together sin2 𝜃 − sin2 𝜃 cos 2 𝜃 cos 2 𝜃 O Factor sin2 𝜃 1 − cos 2 𝜃 cos 2 𝜃 O By ID 1 1 − cos2 𝜃 = sin2 𝜃 sin2 𝜃 sin2 𝜃 cos 2 𝜃 O Rearrange sin2 𝜃 2 sin 𝜃 cos 2 𝜃 2 O By tan 𝜃 = sin2 𝜃 cos2 𝜃 sin2 𝜃 tan2 𝜃 2 2 cos 𝑥 − cos 𝑥 − 1 sec 𝑥 + 2 = cos 𝑥 − 1 sec 𝑥 O LET cos 𝑥 = 𝑢 2𝑢2 − 𝑢 − 1 𝑢−1 O Factor (𝑢 − 1)(2𝑢 + 1) 𝑢−1 O Simplify 2𝑢 + 1 2 cos 𝑥 + 1 O You want SECANT O By reciprocals 1 2 +1 sec 𝑥 2 +1 sec 𝑥 O Common denominator 2 sec 𝑥 + sec 𝑥 sec 𝑥 2 + sec 𝑥 sec 𝑥 Practice, Practice, Practice! O Visions 2 O P132-134: #8, 9, 11, 14, 18 O P140: #4 O Identities_practice jpgs 1-5