7.4PureStretch

Solving Trigonometric Equations

Harder trig equations require an identity — an addition or double-angle formula — to reduce them to a single trig function before solving. Always check the interval.

28 min Video by Zeeshan Zamurred Trigonometry and Modelling

Edexcel A level Maths: 7.4 Solving Trigonometric EquationsWatch the full walkthrough before the notes below.

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What you'll be able to do

Use double-angle formulae to simplify equations
Reduce to a single trig function
Solve the resulting quadratic or linear equation
List all solutions in the interval

Strategy

Replace double or compound angles using an identity so the whole equation is in one function and one angle. Then solve as a standard equation and apply the interval.

1Use

cos 2 θ = 1 - 2 sin^{2} θ

1 - 2 sin^{2} θ + sin θ = 0

2 sin^{2} θ - sin θ - 1 = 0 \Rightarrow (2 sin θ + 1) (sin θ - 1) = 0

sin θ = - \frac{1}{2}

sin θ = 1

Answer

θ = \frac{π}{2}, \frac{7 π}{6}, \frac{11 π}{6}

Tip — Choose the cos 2θ form that leaves only the function already in the equation.

Formula recap

cos 2 θ = 1 - 2 sin^{2} θ

Reduce to sine.

cos 2 θ = 2 cos^{2} θ - 1

Reduce to cosine.

sin 2 θ = 2 sin θ cos θ

Factor a common term.

Common mistakes to avoid

Dividing through by sin θ and losing solutions.

Factor instead — dividing can drop valid roots.

Forgetting solutions in the interval.

Use symmetry and period to list every solution.

Key takeaways

Use an identity to reduce to one function and one angle.
Form a quadratic and factor (don’t divide out).
List all solutions in the given interval.

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