All subjects · Mathematics · WMA13 (P3)

The R addition form

Any expression a cos θ + b sin θ can be written as a single wave, R cos(θ - α) or R sin(θ + α). This makes equations, maximums and minimums easy.

Finding R and α

For a sin θ + b cos θ = R sin(θ + α): expand the right side to R sin θ cos α + R cos θ sin α and compare:

R cos α = a, R sin α = b.

Then R = sqrt(a² + b²) and tan α = b/a, with α acute when a and b are positive.

Which form?

Choose the form whose expansion matches the signs:

a sin θ + b cos θ: R sin(θ + α)
a sin θ - b cos θ: R sin(θ - α)
a cos θ + b sin θ: R cos(θ - α)
a cos θ - b sin θ: R cos(θ + α)

Uses

Solve a cos θ + b sin θ = c: write R cos(θ - α) = c, find θ - α in the extended interval, then add α.
Maximum value R, minimum -R.
For a fraction such as 12/(8 + R sin(θ + α)), the maximum comes from the minimum of the bottom.

Worked example

Write 3 sin x + 4 cos x in the form R sin(x + α), with 0 < α < 90°.

R cos α = 3 and R sin α = 4
R = sqrt(9 + 16) = 5
tan α = 4/3, so α = 53.13°

Answer: 5 sin(x + 53.13°)

Key idea: R = sqrt(a² + b²). Compare coefficients to get R cos α and R sin α, then tan α = (R sin α)/(R cos α). The maximum is R and the minimum is -R.

Check yourself

5cos x - 12sin x = R cos(x + α), where R > 0. Find R.
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13. R = sqrt(25 + 144) = 13.
sqrt(3) sin x + cos x = R sin(x + α), with 0 < α < 90°. Find α in degrees.
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30 °. R cos α = sqrt(3) and R sin α = 1, so tan α = 1/sqrt(3) and α = 30°.
7cos θ + 24sin θ = R cos(θ - α), with 0 < α < 90°. Find α to 2 decimal places.
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73.74 °. R cos α = 7 and R sin α = 24, so tan α = 24/7 and α = 73.74°.
Find the maximum value of 3sin x + 4cos x + 2.
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7. 3sin x + 4cos x = 5sin(x + α), which has maximum 5. Adding 2 gives 7.
Find the maximum value of 12/(8 + 3sin x - 4cos x).
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4. 3sin x - 4cos x = 5sin(x - α) has minimum -5. The fraction is largest when the bottom is smallest: 12/(8 - 5) = 4.
Solve 3sin x + 4cos x = 2 for 0° < x < 180°. Give x to 1 decimal place.
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103.3 °. 5sin(x + 53.13°) = 2, so sin(x + α) = 0.4. x + α lies between 53.13° and 233.13°, so x + α = 180° - 23.58° = 156.42°, and x = 103.3°.
When writing a cos θ + b sin θ as R cos(θ - α), which equations do you compare?
- R cos α = a, R sin α = b
- R sin α = a, R cos α = b
- R cos α = b, R sin α = -a
- R = a + b, tan α = a/b
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R cos α = a, R sin α = b. R cos(θ - α) = R cos θ cos α + R sin θ sin α. Matching gives R cos α = a and R sin α = b.

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Last updated 7 November 2026. Spotted a mistake? Tell us.