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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 Video by Zeeshan Zamurred Radians

Edexcel A Level Maths: Chapter 5 Radians (playlist)Watch the full walkthrough before the notes below.

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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

Working in radians

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

sin θ, cos θ : period 2 π; tan θ : period π

Periods in radians.

Exact values

Know the exact-value angles in radians: $sin \frac{π}{6} = \frac{1}{2}$ , $cos \frac{π}{4} = \frac{2}{2}$ , $tan \frac{π}{3} = 3$ , and so on.

1Principal value

θ = \frac{π}{6}

2Second solution (sin positive in Q2):

π - \frac{π}{6} = \frac{5 π}{6}

Answer

θ = \frac{π}{6}, \frac{5 π}{6}

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

Multiple and shifted angles

For $sin (2 θ) = k$ on $0 \leq θ < 2 π$ , first solve for $2 θ$ on $0 \leq 2 θ < 4 π$ , then divide each solution by 2.

Formula recap

sin, cos : period 2 π

Repeat every 2π.

tan : period π

Repeat every π.

sin θ > 0 in Q 1, Q 2

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.

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