Question

In this warm-up activity, you will use your knowledge from the previous lesson on compound angle formulas to derive expressions for the double angle formulas.

Derive a general expression for sin(2θ) and cos(2θ). Hint: sin(2θ) = sin(θ + θ), and use the compound angle formula that was introduced in the previous lesson. Be sure to do this for both sin(2θ) and cos (2θ).

Answer 1

The double angle formulas for sin(2θ) and cos(2θ) are derived by applying compound angle formulas. sin(2θ) = 2sinθcosθ and cos(2θ) can be expressed as either 1 - 2sin²θ or 2cos²θ - 1.

Step-by-step explanation:

To derive the expression for sin(2θ), we will utilize the compound angle formula for the sine of a sum: sin(α + β) = sinαcosβ + cosαsinβ. By substituting α and β with θ, we get sin(θ + θ) = sinθcosθ + cosθsinθ, which simplifies to sin(2θ) = 2sinθcosθ.

Similarly, the expression for cos(2θ) derives from the compound angle formula for cosine: cos(α + β) = cosαcosβ - sinαsinβ. When α and β are both θ, we get cos(2θ) = cosθcosθ - sinθsinθ, which results in two different forms: cos(2θ) = 1 - 2sin²θ or cos(2θ) = 2cos²θ - 1, utilizing the Pythagorean identity sin²θ + cos²θ = 1.