Open MathNSWGitHub ↗

Trigonometric equations

Solve trigonometric equations for all solutions in a given domain, including equations requiring Pythagorean identity substitution and equations that reduce to a quadratic in a trigonometric function.

Worked examples

Solving a basic trigonometric equation

Straightforward

Problem

Solve cos⁡θ=12\cos\theta = \dfrac{1}{2} for 0°≤θ≤360°0° \leq \theta \leq 360°.

Solving using a Pythagorean identity substitution

Moderate

Problem

Solve 2cos⁡2θ+5sin⁡θ+1=02\cos^2\theta + 5\sin\theta + 1 = 0 for 0°≤θ≤360°0° \leq \theta \leq 360°.

Solving a quadratic trigonometric equation

Challenging

Problem

Solve 2cos⁡2θ−cos⁡θ−1=02\cos^2\theta - \cos\theta - 1 = 0 for 0≤θ≤2π0 \leq \theta \leq 2\pi.

Practise

Q1·Straightforward
Solve sin⁡θ=12\sin\theta = \dfrac{1}{2} for 0°≤θ≤360°0° \leq \theta \leq 360°. Which list contains all solutions?
Q2·Straightforward
Solve cos⁡θ=0\cos\theta = 0 for 0°≤θ≤360°0° \leq \theta \leq 360°. The smaller solution is 90°90°. What is the larger solution in degrees?
Q3·Straightforward
Solve tan⁡θ=3\tan\theta = \sqrt{3} for 0°≤θ≤360°0° \leq \theta \leq 360°. The smaller solution is 60°60°. What is the larger solution in degrees?
Q4·Straightforward
Solve sin⁡θ=−1\sin\theta = -1 for 0≤θ≤2π0 \leq \theta \leq 2\pi. What is the solution?
Q5·Moderate
Solve 2cos⁡2θ+3sin⁡θ=32\cos^2\theta + 3\sin\theta = 3 for 0°≤θ≤360°0° \leq \theta \leq 360°. What is the largest solution in degrees?
Q6·Moderate
Solve cos⁡2θ=32sin⁡θ\cos^2\theta = \dfrac{3}{2}\sin\theta for 0°≤θ≤360°0° \leq \theta \leq 360°. What is the sum of all solutions in degrees?
Q7·Moderate
Solve 2sin⁡2θ−cos⁡θ−1=02\sin^2\theta - \cos\theta - 1 = 0 for 0°≤θ≤360°0° \leq \theta \leq 360°. What is the sum of all solutions in degrees?
Q8·Moderate
Solve 2cos⁡2θ−sin⁡θ−1=02\cos^2\theta - \sin\theta - 1 = 0 for 0°≤θ≤360°0° \leq \theta \leq 360°. What is the sum of all solutions in degrees?
Q9·Challenging
Solve 2sin⁡2θ−sin⁡θ−1=02\sin^2\theta - \sin\theta - 1 = 0 for 0≤θ≤2π0 \leq \theta \leq 2\pi. Which set contains all solutions?
Q10·Challenging
Solve tan⁡2θ−tan⁡θ=0\tan^2\theta - \tan\theta = 0 for 0≤θ<2π0 \leq \theta < 2\pi. Which set contains all solutions?
Q11·Challenging
Solve 4sin⁡2θ−1=04\sin^2\theta - 1 = 0 for 0≤θ≤2π0 \leq \theta \leq 2\pi. Which set contains all solutions?
Q12·Challenging
Solve cos⁡2θ+cos⁡θ=0\cos^2\theta + \cos\theta = 0 for 0≤θ≤2π0 \leq \theta \leq 2\pi. Which set contains all solutions?