Trigonometry

Key concepts

• sin²θ + cos²θ = 1
• tan θ = sin θ / cos θ
• Compound: sin(A±B), cos(A±B)
• Double: sin 2A = 2 sin A cos A; cos 2A = cos²A − sin²A
• Sine rule: a/sin A = b/sin B = c/sin C
• Cosine rule: a² = b² + c² − 2bc cos A
• Area = ½·ab·sin C

How it works

Decide whether you have a right-angled triangle (use SOH-CAH-TOA), a non-right triangle (use sine or cosine rule), or an identity to prove. For equations, isolate the trig ratio first, then write the general solution and select values in the given interval. When an expression has angles like 180° + x or −x, reduce first. When it has 2x or a sum of angles, expand with the double- or compound-angle identities. When it is an equation, aim for a single ratio equal to a number, or factorise, then find the reference angle and the quadrants.

Step by step

1. Classify the problem

Equation, identity, triangle rule, or graph?

2. State the rule

Write the identity or rule before substituting.

3. Work in degrees

Set your calculator to degrees mode unless told otherwise.

4. Check the interval

List ALL solutions in the given interval.

Worked examples

Solve sin x = 0.5 for x ∈ [0°, 360°].

1. Reference angle = 30°
2. x = 30° or x = 180° − 30° = 150°

Answer: x = 30° or 150°

Prove: (1 − cos 2A) / sin 2A = tan A

1. LHS = (1 − (1 − 2sin²A)) / (2 sin A cos A)
2. = 2 sin²A / (2 sin A cos A)
3. = sin A / cos A = tan A ✓

Answer: Identity proved

Find the general solution of cos 2x = 0.5

1. The reference angle is 60°, so 2x = ±60° + k·360°
2. Divide every term by 2

Answer: x = ±30° + k·180°, k ∈ ℤ

If sin 28° = p, express cos 62° and sin 56° in terms of p.

1. cos 62° = sin(90° − 62°) = sin 28° = p
2. 28° is acute, so cos 28° = √(1 − p²)
3. sin 56° = 2 sin 28° cos 28°

Answer: cos 62° = p and sin 56° = 2p√(1 − p²)

Common mistakes

• Forgetting the second quadrant solution for sine equations
• Mixing up sine and cosine rule conditions
• Cancelling sin or cos terms incorrectly in identities

Frequently asked questions

Is trig in Paper 1?

Trig is mostly Paper 2. Paper 1 may include logs and exponents — different topic.

Are 3D trig problems still tested?

Yes, expect at least one 3D problem in Paper 2.

What is a reference angle?

The acute angle between the terminal side of an angle and the x-axis. You use it with the CAST diagram to find every angle in an interval with the same ratio value.