How do you derive the double-angle formulas for sine, cosine, and tangent, and how can these be applied to solve identities and equations in non-right triangles?
Answer 1
The double-angle formulas are derived from the sum formulas for sine and cosine. For sine: sin(2θ) = 2sin(θ)cos(θ). For cosine: cos(2θ) = cos²(θ) – sin²(θ) or cos(2θ) = 2cos²(θ) – 1 or cos(2θ) = 1 – 2sin²(θ). For tangent: tan(2θ) = 2tan(θ) / (1 – tan²(θ)). These formulas are crucial in solving trigonometric identities and equations in non-right triangles, particularly in the Law of Sines and Law of Cosines, which are used to find unknown sides and angles.
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