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A2.F.TF.8 2011 Domain: Trigonometric Functions Cluster: Prove and apply trigonometric identities Standards: Prove the Pythagorean identity sin2(θ) + cos2(θ) = 1 and use it to find sin(θ), cos(θ), or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle. Essential Questions What is a trigonometric identity? Why are some identities called trigonometric identities? Content Statements Graph trigonometric functions. Enduring Understandings Establish a link between the different forms of the Pythagorean Theorem Geometry: a2 + b2 = c2 Algebra 2: a2 + b2 = c2 Unit circle: x2 + y2 = r2 Identity: sin2(θ) + cos2(θ) = 1 Verify trigonometric identities. Apply/model trigonometric identities to real life situations. Assessments Rewrite each expression in terms of a single trigonometric function a. tan θ cot θ Answer: 1 b. Answer: 1/sin θ sin 1 cos 2 Activities, Investigation, and Student Experiences Given that sin θ = 4/5 and π/2 < θ < π, find the values of the five remaining trigonometric functions of θ. Simplify the expression tan (π/2 – θ)sin θ sec2 1 Verify the identity sin 2 2 sec The length of a shadow is given by the formula h sin(90 ) s sin o Simplify this expression Use the Pythagorean identity sin2 θ + cos2 θ = 1 to derive the other Pythagorean identities, 1 + tan2 θ = sec2 θ and 1 + cot2 θ = csc2 θ A2.F.TF.8 Which expression is equivalent to sec θ sin θ? a. sin θ b. cos θ c. csc θ d. tan θ ** Verify the trigonometric identity sec 1 2 sec 2 sin 2 Answer sec 2 sec 2 1 sec 1 1 sec 1 cos 2 sin 2 2 2 Equipment Needed: Teacher Resources: http://www.khanacademy.org/video/trigonometricidentities?playlist=Trigonometry 2011 A2.F.TF.8 Holt McDougal, Algebra 2, (2012) 2011