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6. APC Lesson 26  Essential Question: What is the cosine double angle identity?  Standard: Prove and apply trigonometric identities MCC9 ‐ 12.F.TF.9 (+) Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.  Identity memory quiz: Wednesday

7.  Proof:

8. Cos( α +- β) = cos(α )cos( β ) -+ sin( α )sin( β ) Sum/Difference Identity for cosine

9. Well then what is cos(2Ѳ)?

10. What are three ways to write the double angle identity for cosine? 1. cos(2 Ѳ) = cos 2 (Ѳ) – sin 2 (Ѳ) 2. cos(2 Ѳ) = 1 - 2 sin 2 (Ѳ) 3. cos(2 Ѳ) = 2 cos 2 (Ѳ) – 1 _____________________________ Proof of #2 sin 2 (Ѳ) + cos 2 (Ѳ) = 1 cos 2 (Ѳ) = 1 - sin 2 (Ѳ) cos 2 (Ѳ)- sin 2 (Ѳ) = 1 - sin 2 (Ѳ) - sin 2 (Ѳ) Cos(2 Ѳ) = 1 - 2 sin 2 (Ѳ) Q.E.D

11. Proof of # 3 sin 2 (Ѳ) + cos 2 (Ѳ) = 1 sin 2 (Ѳ) = 1 - cos 2 (Ѳ) cos 2 (Ѳ) - sin 2 (Ѳ) = cos 2 (Ѳ) –(1- cos 2 (Ѳ) ) Cos(2 Ѳ) = 2 cos 2 (Ѳ) - 1 Q.E.D

12. [1/sec(x)+1/csc(x) ][1/sec(x) – 1/csc(x)] = cos(2x)

13.  Tan(2 Ѳ ) = sin(2 Ѳ ) / cos(2 Ѳ )

14. G IVEN SIN ( X ) = -2/3 AND X IS IN QIII F IND THE FOLLOWING : Cos(x) = Sin(2x) = Tan(2x) =