Back to course
10.4PureStretch

Simple Trigonometric Equations

Equations like sin⁡θ=0.5\sin\theta = 0.5 usually have several solutions in a given range. The calculator gives one; the CAST diagram or graph finds the rest. Getting every solution in the range is the whole skill here.

30 min Trigonometric Identities and Equations
Edexcel AS Level Maths: 10.4 Solving Trigonometric EquationsCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Find the principal value from a calculator
  • Use CAST or the graph to find all solutions in a range
  • Solve equations involving sin, cos and tan
  • Work within a stated interval
1

The principal value

A calculator’s sin⁡−1\sin^{-1}, cos⁡−1\cos^{-1} or tan⁡−1\tan^{-1} returns one angle — the principal value\textbf{principal value}. This is your starting point, not the full answer.

Example 1Find the principal value for sin⁡θ=0.5\sin\theta = 0.5.
1sin⁡−1(0.5)=30∘\sin^{-1}(0.5) = 30^\circ.
Answerθ=30∘\theta = 30^\circ
2

Finding all solutions

Use the symmetry of the graph (or CAST) to generate the other solutions in the range. For sine within 0∘0^\circ–360∘360^\circ, the second solution is 180∘−θ180^\circ - \theta; for cosine it is 360∘−θ360^\circ - \theta; tangent repeats every 180∘180^\circ.

sin⁡: θ and 180∘−θ\sin: \ \theta \text{ and } 180^\circ - \theta
Within one revolution.
cos⁡: θ and 360∘−θ\cos: \ \theta \text{ and } 360^\circ - \theta
Within one revolution.
tan⁡: θ and θ+180∘\tan: \ \theta \text{ and } \theta + 180^\circ
Period 180°.
Example 1Solve sin⁡θ=0.5\sin\theta = 0.5 for 0∘≤θ≤360∘0^\circ \le \theta \le 360^\circ.
1Principal value 30∘30^\circ.
2Second solution 180∘−30∘=150∘180^\circ - 30^\circ = 150^\circ.
Answerθ=30∘,150∘\theta = 30^\circ, 150^\circ

Tip — Sketch the curve and the horizontal line y = k: every crossing in the range is a solution.

3

Negative and other cases

If the calculator gives a value outside the range, add or subtract 360∘360^\circ (or 180∘180^\circ for tan) to bring solutions into range. A negative right-hand side just lands the solutions in different quadrants — CAST still guides you.

Formula recap

sin⁡θ=k⇒θ, 180∘−θ\sin\theta = k \Rightarrow \theta, \ 180^\circ - \theta
Sine in 0°–360°.
cos⁡θ=k⇒θ, 360∘−θ\cos\theta = k \Rightarrow \theta, \ 360^\circ - \theta
Cosine in 0°–360°.
tan⁡θ=k⇒θ, θ+180∘\tan\theta = k \Rightarrow \theta, \ \theta + 180^\circ
Tangent (period 180°).

Common mistakes to avoid

Giving only the calculator’s principal value.
Find every solution in the stated range using symmetry or CAST.
Forgetting to keep solutions within the given interval.
Discard any outside the range; add/subtract a period to bring others in.

Key takeaways

  • The calculator gives one principal value to start from.
  • sin: θ and 180° − θ; cos: θ and 360° − θ; tan: θ and θ + 180°.
  • A quick sketch shows how many solutions lie in the range.
  • Keep only the solutions within the stated interval.

Test yourself

Answer a few questions, then review anything you missed.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 6 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.