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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

SIN (2Ѳ) = 2SIN(Ѳ) COS(Ѳ)

Proof:

Cos(α +- β) = cos(α)cos(β) -+ sin(α)sin(β)

Sum/Difference Identity for cosine

Well then what is cos(2Ѳ)?

What are three ways to write the double angle identity for cosine?

1. cos(2 Ѳ) = cos2(Ѳ) – sin 2(Ѳ) 2. cos(2 Ѳ) = 1 - 2 sin 2(Ѳ) 3. cos(2 Ѳ) = 2 cos2(Ѳ) – 1 _____________________________

Proof of #2sin 2(Ѳ) + cos2(Ѳ) = 1 cos2(Ѳ) = 1 - sin 2(Ѳ) cos2(Ѳ)- sin 2(Ѳ) = 1 - sin 2(Ѳ) - sin 2(Ѳ) Cos(2 Ѳ) = 1 - 2 sin 2(Ѳ) Q.E.D

Proof of # 3

• sin 2(Ѳ) + cos2(Ѳ) = 1 • sin 2(Ѳ) = 1 - cos2(Ѳ) • cos2(Ѳ) - sin 2(Ѳ) = cos2(Ѳ) –(1- cos2(Ѳ) ) • Cos(2 Ѳ) = 2 cos2(Ѳ) - 1 Q.E.D

Prove the following[1/sec(x)+1/csc(x) ][1/sec(x) – 1/csc(x)] = cos(2x)

Note to remember

Tan(2Ѳ) = sin(2Ѳ) / cos(2Ѳ)

GIVEN SIN(X) = -2/3 AND X IS IN QIII FIND THE FOLLOWING:

Cos(x) =

Sin(2x) =

Tan(2x) =