Back to course
5.4PureStretch

Solving Trigonometric Equations

Trig equations are solved exactly as in Year 1, but with the interval and answers expressed in radians. Use exact values and symmetry of the trig graphs to find every solution in the given range.

30 min Radians
Edexcel A Level Maths: Chapter 5 Radians (playlist)Curated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Solve sin, cos, tan equations in radians
  • Find all solutions in a given interval
  • Use the CAST/graph symmetry in radians
  • Handle multiples and shifts of the angle
1

Working in radians

Find the principal value, then use the period and symmetry to get all solutions in the interval. Period of sin⁡\sin and cos⁡\cos is 2π2\pi; period of tan⁡\tan is π\pi.

sin⁡θ, cos⁡θ:period 2π;tan⁡θ:period π\sin\theta,\ \cos\theta: \text{period } 2\pi; \quad \tan\theta: \text{period } \pi
Periods in radians.
2

Exact values

Know the exact-value angles in radians: sin⁡π6=12\sin\tfrac{\pi}{6} = \tfrac12, cos⁡π4=22\cos\tfrac{\pi}{4} = \tfrac{\sqrt2}{2}, tan⁡π3=3\tan\tfrac{\pi}{3} = \sqrt3, and so on.

Example 1Solve sin⁡θ=12\sin\theta = \tfrac12 for 0≤θ<2π0 \le \theta < 2\pi.
1Principal value θ=π6\theta = \tfrac{\pi}{6}.
2Second solution (sin positive in Q2): π−π6=5π6\pi - \tfrac{\pi}{6} = \tfrac{5\pi}{6}.
Answer
θ=π6,5π6\theta = \tfrac{\pi}{6}, \tfrac{5\pi}{6}

Tip — Watch the interval: a multiple angle like 2θ needs a widened interval before you solve.

3

Multiple and shifted angles

For sin⁡(2θ)=k\sin(2\theta) = k on 0≤θ<2π0 \le \theta < 2\pi, first solve for 2θ2\theta on 0≤2θ<4π0 \le 2\theta < 4\pi, then divide each solution by 2.

Formula recap

sin⁡,cos⁡:period 2π\sin, \cos: \text{period } 2\pi
Repeat every 2π.
tan⁡:period π\tan: \text{period } \pi
Repeat every π.
sin⁡θ>0 in Q1,Q2\sin\theta>0 \text{ in } Q1,Q2
Symmetry: π − θ.

Common mistakes to avoid

Giving only the principal value.
Use symmetry/period to find every solution in the interval.
Not widening the interval for a multiple angle.
For 2θ on [0, 2π), solve 2θ on [0, 4π) first.

Key takeaways

  • Solve as in Year 1 but in radians.
  • sin, cos period 2π; tan period π.
  • Widen the interval for multiple angles, then divide back.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. The period of sin⁡θ\sin\theta in radians is:

    • 2π2\pi
    • π\pi
    • π2\tfrac{\pi}{2}
    • 360360
    Show answer and explanation

    Answer: 2π2\pi

    Full circle.

  2. The period of tan⁡θ\tan\theta is:

    • π\pi
    • 2π2\pi
    • π2\tfrac{\pi}{2}
    • 4π4\pi
    Show answer and explanation

    Answer: π\pi

    tan repeats every π.

  3. Solutions of sin⁡θ=12\sin\theta = \tfrac12 on [0,2π)[0, 2\pi):

    • π6,5π6\tfrac{\pi}{6}, \tfrac{5\pi}{6}
    • π6\tfrac{\pi}{6}
    • π3,2π3\tfrac{\pi}{3}, \tfrac{2\pi}{3}
    • π6,7π6\tfrac{\pi}{6}, \tfrac{7\pi}{6}
    Show answer and explanation

    Answer: π6,5π6\tfrac{\pi}{6}, \tfrac{5\pi}{6}

    sin positive in Q1, Q2.

  4. For sin⁡2θ=k\sin 2\theta = k on [0,2π)[0,2\pi), first solve 2θ2\theta on:

    • [0,4π)[0, 4\pi)
    • [0,2π)[0, 2\pi)
    • [0,π)[0, \pi)
    • [0,8π)[0, 8\pi)
    Show answer and explanation

    Answer: [0,4π)[0, 4\pi)

    Double the interval.

  5. tan⁡π3\tan\tfrac{\pi}{3} equals:

    • 3\sqrt3
    • 13\tfrac{1}{\sqrt3}
    • 11
    • 32\tfrac{\sqrt3}{2}
    Show answer and explanation

    Answer: 3\sqrt3

    π/3 = 60°.

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 5 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.