General solution

Definition

Because trigonometric functions are periodic, a trig equation has infinitely many solutions. The general solution adds k·360° for sin and cos, or k·180° for tan, with k ∈ ℤ.

Worked example

sinθ = 0,5 gives θ = 30° + k·360° or θ = 150° + k·360°, k ∈ ℤ.

cosθ = −0,5: the reference angle is 60°, and cos is negative in quadrants II and III, so θ = 120° + k·360° or θ = 240° + k·360°, k ∈ ℤ.

Where it appears in the exam

Paper 2 asks for the general solution of a trig equation, often after factorising or using an identity, and sometimes for the solutions in a given interval.

Step-by-step method

1. Isolate the trig ratio.
2. Find the reference angle from the positive value.
3. Use CAST to find the angles in the correct quadrants.
4. Add k·360° for sin and cos, or k·180° for tan, with k ∈ ℤ.

Common mistake

Leaving out 'k ∈ ℤ', which loses a mark.

Related trigonometry terms

• Reduction formula
• Cosine rule

Frequently asked questions

What does general solution mean in Grade 12 Maths?

Because trigonometric functions are periodic, a trig equation has infinitely many solutions. The general solution adds k·360° for sin and cos, or k·180° for tan, with k ∈ ℤ.

Which NSC paper tests general solution?

General solution belongs to the Trigonometry section of CAPS Grade 12 Mathematics, which is examined in Paper 2.

How do I practise questions involving general solution?

Snap a question that uses general solution into the Snap&Learn AI solver, or work through the trigonometry topic guide and NSC past papers.