Question 1

What are the double angle formulas?

Accepted Answer

sin(2α) = 2sinα·cosα, cos(2α) = cos²α - sin²α = 2cos²α - 1 = 1 - 2sin²α, and tan(2α) = 2tanα/(1 - tan²α). These express functions of 2α in terms of α.

Question 2

What are the half angle formulas?

Accepted Answer

sin(α/2) = ±√[(1-cosα)/2], cos(α/2) = ±√[(1+cosα)/2], tan(α/2) = (1-cosα)/sinα = sinα/(1+cosα). The sign depends on the quadrant of α/2.

Question 3

What are sum-to-product formulas?

Accepted Answer

sin A + sin B = 2sin[(A+B)/2]cos[(A-B)/2], sin A - sin B = 2cos[(A+B)/2]sin[(A-B)/2], cos A + cos B = 2cos[(A+B)/2]cos[(A-B)/2], cos A - cos B = -2sin[(A+B)/2]sin[(A-B)/2].

Question 4

What are product-to-sum formulas?

Accepted Answer

sin A cos B = ½[sin(A+B) + sin(A-B)], cos A sin B = ½[sin(A+B) - sin(A-B)], cos A cos B = ½[cos(A+B) + cos(A-B)], sin A sin B = -½[cos(A+B) - cos(A-B)].

Question 5

When should I use these identities?

Accepted Answer

Double angle formulas simplify expressions with 2α. Half angle formulas help find exact values of special angles. Sum/product formulas are useful for simplifying sums into products (easier to factor) or products into sums (easier to integrate).

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