10.4PureStretch

Simple Trigonometric Equations

Equations like $sin θ = 0.5$ 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 Video by Zeeshan Zamurred Trigonometric Identities and Equations

Edexcel AS Level Maths: 10.4 Solving Trigonometric EquationsWatch the full walkthrough before the notes below.

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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

The principal value

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

sin^{- 1} (0.5) = 3 0^{\circ}

Answer

θ = 3 0^{\circ}

Finding all solutions

Use the symmetry of the graph (or CAST) to generate the other solutions in the range. For sine within $0^{\circ}$ – $36 0^{\circ}$ , the second solution is $18 0^{\circ} - θ$ ; for cosine it is $36 0^{\circ} - θ$ ; tangent repeats every $18 0^{\circ}$ .

sin : θ and 18 0^{\circ} - θ

Within one revolution.

cos : θ and 36 0^{\circ} - θ

Within one revolution.

tan : θ and θ + 18 0^{\circ}

Period 180°.

1Principal value

3 0^{\circ}

2Second solution

18 0^{\circ} - 3 0^{\circ} = 15 0^{\circ}

Answer

θ = 3 0^{\circ}, 15 0^{\circ}

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

Negative and other cases

If the calculator gives a value outside the range, add or subtract $36 0^{\circ}$ (or $18 0^{\circ}$ 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 \Rightarrow θ, 18 0^{\circ} - θ

Sine in 0°–360°.

cos θ = k \Rightarrow θ, 36 0^{\circ} - θ

Cosine in 0°–360°.

tan θ = k \Rightarrow θ, θ + 18 0^{\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

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