Compound Angle Formulae

The compound angle (or addition) formulae express sin(A ± B), cos(A ± B) and tan(A ± B) in terms of sin and cos of A and B individually. These are given in the Edexcel formula booklet and are essential for A-Level (9MA0).

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The Six Compound Angle Formulae

Sine

sin(A + B) = sinA cosB + cosA sinB

sin(A - B) = sinA cosB - cosA sinB

Cosine

cos(A + B) = cosA cosB - sinA sinB

cos(A - B) = cosA cosB + sinA sinB

Tangent

tan(A + B) = (tanA + tanB) / (1 - tanA tanB)

tan(A - B) = (tanA - tanB) / (1 + tanA tanB)

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Using the Formulae to Find Exact Values

Worked Example 1

Find the exact value of sin(75 degrees) = sin(pi/12 x 5) = sin(45 + 30).

sin(75) = sin(45 + 30) = sin45 cos30 + cos45 sin30

= (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2)

= sqrt(6)/4 + sqrt(2)/4

= (sqrt(6) + sqrt(2))/4

Worked Example 2

Find the exact value of cos(15 degrees) = cos(45 - 30).

cos(15) = cos45 cos30 + sin45 sin30

= (sqrt(2)/2)(sqrt(3)/2) + (sqrt(2)/2)(1/2)

= sqrt(6)/4 + sqrt(2)/4