Grade 12 · Pure Mathematics 2 · Cambridge A-Level 9709 · Age 17–18
Compound angle formulae are the foundation for solving harder trigonometric equations, proving identities, and applying trigonometry in calculus. Mastering these formulae unlocks the entire A-Level trig toolkit.
These three formulae — and their minus-angle counterparts — generate all double angle results and enable the R·sin(θ+α) method for solving equations.
sin/cos/tan of A±B — the core formulae for all advanced trig
sin(2A), three forms of cos(2A), and tan(2A)
Combine a·sinθ + b·cosθ into a single sinusoidal wave
Use identities to reduce equations to simpler forms
Work from one side using compound angle results
Calculus, wave analysis, and exam-style problems
These four formulae are the starting point for everything in this topic. Learn to recognise both the + and − versions, noting how the signs differ between sin and cos.
These formulae let us find exact trig values for angles not in the standard table by splitting them into 30°, 45°, 60° combinations.
Setting B = A in the compound angle formulae gives the double angle formulae. These are among the most important results at A-Level.
Any expression of the form a·sinθ + b·cosθ can be written as a single sinusoidal wave R·sin(θ + α). This is essential for finding maximum/minimum values and solving equations.
The key strategy is to use identities to express an equation in terms of a single trig ratio, then solve as usual.
For cos, the sign flips relative to the angle. cos(A+B) uses minus, cos(A−B) uses plus. Contrast with sin where the sign matches.
R is the square root of the sum of squares — always take the positive square root.
Choose the form of cos(2A) that eliminates the trig ratio you don't want.
The denominators of tan(A+B) and tan(A−B) have opposite signs to what you'd expect. Memorise this carefully.
Sine equations always have two solutions per period. Use the principal value and the supplementary angle, then adjust for range.
Doubling the angle is not the same as doubling the ratio. The formula requires both sinA and cosA.
Always check every solution against the domain restriction at the end.
| Formula | Expression | Notes |
|---|---|---|
| sin(A+B) | sinA cosB + cosA sinB | Signs match the angle |
| sin(A−B) | sinA cosB − cosA sinB | Signs match the angle |
| cos(A+B) | cosA cosB − sinA sinB | Sign flips! |
| cos(A−B) | cosA cosB + sinA sinB | Sign flips! |
| tan(A+B) | (tanA + tanB)/(1 − tanA tanB) | Denominator: minus |
| tan(A−B) | (tanA − tanB)/(1 + tanA tanB) | Denominator: plus |
| sin(2A) | 2sinA cosA | From sin(A+A) |
| cos(2A) — Form 1 | cos²A − sin²A | Direct substitution |
| cos(2A) — Form 2 | 2cos²A − 1 | Use when eliminating sinA |
| cos(2A) — Form 3 | 1 − 2sin²A | Use when eliminating cosA |
| tan(2A) | 2tanA/(1 − tan²A) | From tan(A+A) |
| R·sin(θ+α) | R = √(a²+b²), tanα = b/a | a sinθ + b cosθ form |
| R·cos(θ−α) | R = √(a²+b²), tanα = b/a | a cosθ + b sinθ form |
| Max of R·sin(θ+α) | R | When sin(θ+α) = 1 |
| Min of R·sin(θ+α) | −R | When sin(θ+α) = −1 |
| sin²A (rearranged) | (1 − cos2A)/2 | Useful in integration |
| cos²A (rearranged) | (1 + cos2A)/2 | Useful in integration |
Adjust a and b to see how the combined wave changes. The blue line is a·sinθ + b·cosθ. The red dashed line is R·sin(θ+α). They are identical — this proves the R-form works!
(a) Express 8sinθ + 6cosθ in the form R sin(θ + α), where R > 0 and 0° < α < 90°. State the exact value of R and give α to 2 decimal places. [3]
(b) Hence find the maximum value of 8sinθ + 6cosθ and the smallest positive θ at which it occurs. [2]
Solve the equation 5sinx − 12cosx = 6.5 for 0° ≤ x ≤ 360°, giving solutions to 1 decimal place. [6]
(a) Show that cos(2θ) + 3cosθ + 2 = 0 can be written as (2cosθ + 1)(cosθ + 1) = 0. [2]
(b) Hence solve cos(2θ) + 3cosθ + 2 = 0 for 0° ≤ θ ≤ 360°. [3]
Prove the identity: (sin2A + sin A) / (cos2A + cosA + 1) = tanA [4]
f(θ) = 3sinθ + 4cosθ
(a) Write f(θ) in the form R sin(θ + α) where R > 0 and 0 < α < π/2. Give R exactly and α to 3 significant figures. [3]
(b) Hence find all solutions of f(θ) = 2 for 0 ≤ θ ≤ 2π. Give answers to 3 significant figures. [3]
Given that sin(A+B) = 2sin(A−B), show that tanA = 3tanB. [5]
(a) Find the exact value of sin(π/12). [3]
(b) Hence, or otherwise, find the exact value of sin²(π/12). [1]
(c) Find ∫₀^(π/12) sin²(x) dx, leaving your answer in exact form. [2]
Solve the equation sin(2θ) = sinθ for 0 ≤ θ ≤ 2π. [4]
The function f is defined by f(x) = 5sinx − 12cosx for x ∈ ℝ.
(a) Express f(x) in the form R sin(x − α), where R > 0 and 0 < α < π/2. [3]
(b) State the range of f. [1]
(c) Solve f(x) = 6 for 0 ≤ x ≤ 2π. [3]
Prove that (1 − cos2θ)/(sin2θ) ≡ tanθ. [3]
Solve the equation 4cos(2x) + 2cosx = 1 for 0° ≤ x ≤ 360°. [5]
Given that sin(A+B)/cos(A−B) = 3/5, find the value of (tanA + tanB)/(1 + tanA tanB). [3]
f(θ) = 2sinθ + 3cosθ. Express f(θ) in the form R sin(θ+α). Find the minimum value of [f(θ)]² and the corresponding value of θ in [0, 2π]. [6]