Harmonic Form

The harmonic form rewrites expressions like a sin(theta) + b cos(theta) as a single sine or cosine function. This technique is essential for solving certain equations and finding maximum/minimum values at A-Level (9MA0).

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The Harmonic Forms

Any expression of the form a sin(theta) + b cos(theta) can be written as:

R sin(theta + alpha) or R cos(theta - alpha)

where:

R = sqrt(a² + b²)

tan(alpha) = b/a (for R sin(theta + alpha) form)

or equivalently a sin(theta) + b cos(theta) = R sin(theta + alpha) where R > 0 and 0 < alpha < pi/2.

Similarly, a sin(theta) - b cos(theta) = R sin(theta - alpha).

And a cos(theta) + b sin(theta) = R cos(theta - alpha).

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Finding R and alpha

Worked Example 1

Write 3sin(x) + 4cos(x) in the form R sin(x + alpha).

Expand R sin(x + alpha) = R sinx cos(alpha) + R cosx sin(alpha)

Comparing coefficients:

• R cos(alpha) = 3 ... (1)
• R sin(alpha) = 4 ... (2)

R = sqrt(3² + 4²) = sqrt(9 + 16) = sqrt(25) = 5

tan(alpha) = 4/3, so alpha = arctan(4/3) ≈ 0.9273 radians (or 53.13 degrees)