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Name: _________________________________________________________ Date: ________________ Per: _____ LC Math 2 Adv β Deriving a Trigonometric Identity (LT 12) An identity is an equation that is always true regardless of the value(s) of the variable(s). One such identity in trigonometry is π ππ2 π + π ππ2 π = 1, known as the Pythagorean Identity. We are going to prove this identity. When we prove identities, we work only on the most complex side of the equation and transform it into equivalent forms until we reach an equivalent expression that is exactly the same as the expression on the simpler side. For example, 3(x - 4)-2x = x -12 is an identity. We can show this by transforming the more complex side, 3(x - 4)-2x , to show that one of its equivalent forms is x -12, as shown here: 3(π₯ β 4) β 2π₯ = π₯ β 12 3π₯ β 12 β 2π₯ = 3π₯ β 2π₯ β 12 = π₯ β 12 = π₯ β 12 We will be proving that π ππ2 π + π ππ2 π = 1 is an identity. The left side, π ππ2 π + π ππ2 π, is clearly the more complex side of the identity so we will be working to show that one of its equivalent forms is simply 1. Letβs start with a diagram of a right triangle. 1. Label the side lengths of the triangle a, b, and c. 2. Label one of the acute angles of the triangle π. 3. What relationship must exist between a, b, and c? 4. Express π πππ and πππ π in terms of a, b, and c. sinq = ; cosq = 5. Prove the identity. Take the information from #3 and #4 above as given. π ππ2 π + πππ 2 π = 1 Problem Solving 1. Determine which of the following statements are identities. If a statement is an identity, specify for which values of x the equation is true. If a statement is not an identity, explain your reasoning. a) π πππ₯ = π ππ(π₯ + 360°) b) π‘πππ₯ = 1 c) π πππ₯ = π ππ(βπ₯) d) πππ π₯ = π ππ(90 β π₯) 1 2. In a right triangle with acute angle of measure ΞΈ, π πππ = . Use the Pythagorean identity to determine the value 2 of πππ π. 3. Confirm your solution to #2 using a different method. 7 4. In a right triangle with acute angle of measure ΞΈ, π πππ = 9. Use the Pythagorean identity to determine the value of π‘πππ. π 5. Let π πππ = π, where π, π > 0. Express πππ π and π‘πππ in terms of π and π.