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11.3 Fundamental Trigonometric Identities Objectives: F.TF.8: Prove the Pythagorean identity sin2 θ + cos2 θ = 1 and use it to find sin θ, cos θ, or tan θ and the quadrant of the angle. For the Board: You will be able to use fundamental trigonometric identities to simplify and rewrite expressions and to verify other identities. Bell Work 11.3: Simplify: sin A cos2 A 1. cos A sin A sin A 1 2. tan A tan A sin A Anticipatory Set: x 2 y2 Recall: x + y = r on the unit circle. Therefore: 2 2 1 r r 2 2 x y Applying an exponent rule: 1 r r Recall: cos θ = x/r and sin θ = y/r so the above becomes cos2 θ + sin2 θ = 1 2 2 2 This is known as the Pythagorean Identity. Fundamental Trigonometric Identities Reciprocal Identities 1 csc θ = sin θ sec θ = 1 cos θ cot θ = 1 tan θ Tangent and Cotangent Ratio Identities sin θ cos θ tan θ = cot θ = cos θ sin θ Pythagorean Identities cos2 θ + sin2 θ = 1 1 + tan2 θ = sec2 θ cot2 θ + 1 = csc2 θ Negative-Angle Identities sin(-θ) = - sin θ cos(-θ) = cos θ tan(-θ) = - tan θ Open the book to page 772 and read example 1. Example: Prove each trigonomet4ric identity. sec θ a. tan θ = csc θ = 1 cos θ 1 1 1 cos θ sin θ sin θ 1 sin θ sin θ cos θ 1 cos θ tan θ b. 1 – cot θ = 1 + cot(-θ) 1 1 1 =1 tan (θ) - tan θ 1 1 1 cot θ tan θ White Board Activity: Practice: Prove each trigonometric identity. a. sin θ cot θ = cos θ b. 1 – sec(-θ) = 1 – sec θ cos θ sin θ · = cos θ 1 – sec(-θ) = 1 – (1/cos (-θ)) = 1 – (1/cos θ) = 1 – sec θ sin θ Open the book to page 773 and read example 2. Example: Rewrite each expression in terms of cos θ, and simplify. a. sec θ (1 – sin2 θ) b. sin θ cos θ (tan θ + cot θ) (1/cos θ)(1 – (1 – cos2 θ) sin θ cos θ tan θ + sin θ cos θ cot θ sin θ cos θ (1/cos θ)(1 – 1 + cos2 θ) sin θ cos θ sin θ cos θ cos θ sin θ 2 2 2 (1/cos θ)(cos θ) sin θ + cos θ cos θ 1 – cos2 θ + cos2 θ 1 – 2 cos2 θ White Board Activity: Practice: Rewrite each expression in terms of sin, and simplify. a. cos2/(1 – sin2) b. cot2 Assessment: Question Student pairs. Independent Practice: Text: pg. 775 prob. 1 – 6, 8 – 15. Practice: pg. 775 prob. 17 – 37.