Multiple angles and sum-to-product

Double angles come from A = B in the sum formulas:

• sin 2x = 2 sin x cos x = 2 tan x/(1 + tan²x).
• cos 2x = cos²x – sin²x = 2cos²x – 1 = 1 – 2sin²x = (1 – tan²x)/(1 + tan²x).
• tan 2x = 2 tan x/(1 – tan²x).
• Rearranged: 1 + cos 2x = 2cos²x and 1 – cos 2x = 2sin²x. These are the ones you will use most.

Read them as 'the angle on the left is double the angle on the right'. So sin 6x = 2 sin 3x cos 3x, cos 4x = 2cos²2x – 1, and 1 + cos x = 2cos²(x/2).

Example: cos x = 12/13 with x in quadrant I, so sin x = 5/13. Then sin 2x = 2 × (5/13)(12/13) = 120/169, cos 2x = 2 × 144/169 – 1 = 119/169, and tan 2x = 120/119.

Triple angles: sin 3x = 3 sin x – 4 sin³x, cos 3x = 4 cos³x – 3 cos x, and tan 3x = (3 tan x – tan³x)/(1 – 3 tan²x). Check at x = 30°: sin 90° = 3/2 – 4/8 = 1.

Product → sum (to break a product into a sum):

• 2 sin A cos B = sin(A + B) + sin(A – B)
• 2 cos A cos B = cos(A + B) + cos(A – B)
• 2 sin A sin B = cos(A – B) – cos(A + B)
• Example: 2 sin 3x cos x = sin 4x + sin 2x. If the 2 is missing, multiply and divide by 2.

Sum → product (to turn a sum into a product, often to cancel):

• sin C + sin D = 2 sin((C + D)/2) cos((C – D)/2) and sin C – sin D = 2 cos((C + D)/2) sin((C – D)/2).
• cos C + cos D = 2 cos((C + D)/2) cos((C – D)/2) and cos C – cos D = –2 sin((C + D)/2) sin((C – D)/2): watch this minus sign.
• Example: sin 75° + sin 15° = 2 sin 45° cos 30° = 2 × (1/√2)(√3/2) = √6/2.

Pair, factor, cancel. Prove (sin 7x – sin 3x)/(cos 7x + cos 3x) = tan 2x:

• Numerator: 2 cos 5x sin 2x. Denominator: 2 cos 5x cos 2x.
• Cancel 2 cos 5x: sin 2x/cos 2x = tan 2x.