Trigonometry: Identities and Equations

Radians and exact values

At A-Level, angles are usually measured in radians: π rad = 180°. Arc length s = rθ and sector area = ½r²θ (θ in radians).

Learn the exact values for 0, π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°).

The reciprocal functions

• sec x = 1/cos x, cosec x = 1/sin x, cot x = 1/tan x = cos x/sin x.

Key identities

sin²x + cos²x = 1
tan x = sin x ÷ cos x
1 + tan²x = sec²x
1 + cot²x = cosec²x

Compound and double-angle formulae

sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
tan(A ± B) = (tan A ± tan B) ÷ (1 ∓ tan A tan B)

Double angle (put B = A):

sin 2A = 2 sin A cos A
cos 2A = cos²A − sin²A = 2cos²A − 1 = 1 − 2sin²A

Solving trig equations

1. Rearrange to sin x = k (or cos/tan).

2. Find the principal value with the inverse function.

3. Use the graph/CAST diagram to find all solutions in the given interval (sine and cosine repeat every 360°/2π; tan every 180°/π).

4. If the equation has 2x or x + 30, adjust the interval accordingly and solve for the transformed angle first.

Often you must use an identity to reduce the equation to one trig ratio before solving (e.g. replace sin²x with 1 − cos²x to get a quadratic in cos x).

Rcos/Rsin form

a sin x + b cos x can be written as R sin(x + α) or R cos(x − α), where R = √(a² + b²). This is useful for finding maximum/minimum values and solving equations.

Worked example

Solve 2 sin x = 1 for 0 ≤ x < 360°.

• sin x = 0.5 → principal value x = 30°. Sine is also positive in the second quadrant → x = 180° − 30° = 150°.
• Solutions: x = 30° and 150°. ✓

Common mistakes

• Working in degrees when the question uses radians (or vice versa).
• Finding only one solution — use the graph/CAST diagram for all solutions in range.
• Cancelling sin x from both sides and losing solutions (factorise instead).

Exam tips

• Learn the identities and double-angle formulae — they're given sparingly.
• Adjust the interval when the equation involves 2x, ½x or x + a.
• Use sin²x + cos²x = 1 to turn a mixed equation into a solvable quadratic.

Key facts to remember

• Radians: π = 180°; arc s = rθ, sector ½r²θ; learn exact values.
• Identities: sin²x + cos²x = 1, 1 + tan²x = sec²x; compound and double-angle formulae.
• Solve trig equations by finding the principal value then all solutions in range (use graphs/CAST); R sin(x+α) form for a sin x + b cos x.