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Harder Trigonometric Equations

Harder equations add a twist: a multiple angle like $2 θ$ , or a quadratic in $sin θ$ . The methods are the same once you handle the twist carefully — especially the trap of a changed solution range.

35 min Video by Zeeshan Zamurred Trigonometric Identities and Equations

Edexcel AS Level Maths: 10.5 Harder Trigonometric EquationsWatch the full walkthrough before the notes below.

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What you'll be able to do

Solve equations with a multiple angle (e.g. 2θ)
Adjust the solution range for multiple angles
Solve quadratic trigonometric equations
Use a substitution to simplify

Multiple angles

For an equation in $2 θ$ over $0^{\circ} \leq θ \leq 36 0^{\circ}$ , the angle $2 θ$ runs over $0^{\circ}$ – $72 0^{\circ}$ . Solve for $2 θ$ across the $wider$ range first, then divide every solution by $2$ .

1Let the angle be

2 θ

, range

0^{\circ}

–

72 0^{\circ}

2 θ = 3 0^{\circ}, 15 0^{\circ}, 39 0^{\circ}, 51 0^{\circ}

3Divide by 2.

Answer

θ = 1 5^{\circ}, 7 5^{\circ}, 19 5^{\circ}, 25 5^{\circ}

Tip — Widen the range to match the multiple angle BEFORE solving, or you will miss solutions.

Quadratic trig equations

An equation like $2 sin^{2} θ - sin θ - 1 = 0$ is a quadratic in $sin θ$ . Substitute $s = sin θ$ , solve the quadratic, then solve each resulting simple equation.

1Let

s = sin θ

2 s^{2} - s - 1 = (2 s + 1) (s - 1) = 0

2So

sin θ = 1

sin θ = - \frac{1}{2}

sin θ = 1 \Rightarrow 9 0^{\circ}

;

sin θ = - \frac{1}{2} \Rightarrow 21 0^{\circ}, 33 0^{\circ}

Answer

θ = 9 0^{\circ}, 21 0^{\circ}, 33 0^{\circ}

Using an identity first

Sometimes you must replace $sin^{2} θ$ with $1 - cos^{2} θ$ (or vice versa) so the whole equation is in one ratio, turning it into a solvable quadratic.

Tip — If an equation mixes sin² and cos, use the Pythagorean identity to get everything in terms of one ratio.

Formula recap

for n θ : widen the range by \times n

Multiple angle range.

2 s^{2} - s - 1 = 0, s = sin θ

Quadratic in a ratio.

sin^{2} θ = 1 - cos^{2} θ

Identity to unify ratios.

Common mistakes to avoid

Solving sin 2θ = 0.5 in 0°–360° and dividing — missing half the solutions.

Solve over 0°–720° for 2θ first, then divide each by 2.

Discarding sin θ = 1 because it “only gives one solution”.

That is fine — sin θ = 1 gives θ = 90° in range; include it.

Key takeaways

Multiple angle nθ: widen the solution range ×n, solve, then divide.
Quadratic trig equations: substitute s = sin θ (or cos θ), solve, then unwind.
Use the Pythagorean identity to get a single ratio when needed.

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