Problem 90 - Algebra and Trigonometry

Question

Explain how the double-angle formulas are derived.

Step-by-Step Solution

Answer

The double-angle formulas are derived using the sum of angles formulas for sine and cosine. For sine, \( sin(2a) = 2sin(a)cos(a) \). For cosine, two formulas are derived, \( cos(2a) = cos^2(a) - sin^2(a) \) or \( cos(2a) = 2cos^2(a) - 1 \) or \( cos(2a) = 1 - 2sin^2(a) \). For tangent, \( tan(2a) = 2tan(a)/(1 - tan^2(a)) \).

Step 1: Derivation of double-angle formula for sine

The double-angle formula for sine can be derived by using the trigonometric identity for sine of sum of two angles which is given by: \( sin(a+b) = sin(a)cos(b) + cos(a)sin(b) \). Setting \( a = b \) in the identity, we get: \( sin(2a) = 2sin(a)cos(a) \), which is the double-angle formula for sine.

Step 2: Derivation of double-angle formula for cosine

Similarly, the double-angle formula for cosine can be derived from the cosine of sum of two angles formula: \( cos(a+b) = cos(a)cos(b) - sin(a)sin(b) \). Setting \( a = b \) in this identity, we get: \( cos(2a) = cos^2(a) - sin^2(a) \). We can also rewrite this formula in terms of sine or cosine using Pythagorean identity \( sin^2(a) + cos^2(a) = 1 \), hence \( cos(2a) = 2cos^2(a) - 1 \) or \( cos(2a) = 1 - 2sin^2(a) \). Both are the double-angle formulas for cosine.

Step 3: Derivation of double-angle formula for tangent

To derive the double-angle formula for tangent, we divide the sine double-angle formula by the cosine double-angle formula, and use the identity \( tan(a) = sin(a)/cos(a) \). So, \( tan(2a) = sin(2a)/cos(2a) = 2tan(a)/(1 - tan^2(a)) \). This is the double angle formula for tangent.