Free Trigonometry Solver
Type any trig equation, identity or triangle problem and see the full working — unit circle, double-angle formulas, sine rule, cosine rule and inverse trig included.
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Solving sin 2x = ½ — step by step
Below is one fully worked example plus a short primer so you can see exactly how our AI reasons through a problem.
Example Problem
DEMO- 1
Let θ = 2x and find all θ in [0, 4π]
Since 0 ≤ x ≤ 2π, the substitution θ = 2x ranges over [0, 4π]. We'll solve for θ first, then halve to get x.
- 2
Solve sin θ = 1/2 on its principal range
These are the reference angles in [0, 2π] where sine equals 1/2.
- 3
Extend to [0, 4π] by adding 2π
Sine has period 2π, so add 2π to each root to get the full set in [0, 4π].
- 4
Convert back to x by dividing by 2
- 5
Verify one root
x = π/12 → 2x = π/6 → sin(π/6) = 1/2 ✓
Final Answer
How to solve a trigonometry problem — identities, equations and triangle rules
Trigonometry studies the relationship between angles and sides in triangles, and the periodic behaviour of sine, cosine and tangent. Three settings cover almost everything in Class 10–12, GCSE Higher, A-Level and AP: right-triangle trig (SOH-CAH-TOA, Pythagoras) for acute angles; the unit circle and identities (Pythagorean, double-angle, sum-to-product) for the full domain; and the sine and cosine rules for non-right triangles. Equations like sin 2x = 1/2 are solved by reducing to a principal-range equation, then extending using the periodicity of the function. Identities are proved by manipulating one side until it matches the other, usually via Pythagorean or double-angle identities. Applications include bearings, waves, alternating current, projectile motion and 3D vectors — which is why trigonometry reappears across physics, mechanics and engineering exams, not just maths.
Trigonometry problems to practise
Tap any problem to solve it with full step-by-step working.
- Solve with AI →1.Basic equationClass 10 / GCSEEasy
- Solve with AI →2.Identity proofClass 11 / A-LevelEasy
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3. Solve triangle: a = 7, b = 9, C = 60°. Find c.
Cosine ruleGCSE HigherMedium - Solve with AI →4.Quadratic-in-trigA-Level / APMedium
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5. From a point 30 m from the base of a tower, the angle of elevation to the top is 52°. Find the height.
Right-triangle applicationClass 10Easy
Frequently asked questions
How do I remember the unit-circle values for sin, cos and tan?+
Memorise the five key angles 0, 30°, 45°, 60°, 90° in both degrees and radians. Sine increases from 0 to 1 across these; cosine decreases from 1 to 0; tangent goes 0, 1/√3, 1, √3, undefined. Everything else follows from symmetry across quadrants.
When should I use the sine rule vs the cosine rule?+
Use the sine rule when you have either two angles and any side, or two sides and the angle opposite one of them. Use the cosine rule when you have two sides and the included angle (to find the third side), or three sides (to find an angle).
How do I solve a trig equation with multiple solutions?+
Find the principal-range solution first (usually [0, 2π] or [0°, 360°]), then extend using the period of the function. Sine and cosine have period 2π; tangent has period π. For equations like sin(2x) = ½, expand the interval before solving, then halve at the end.
Does this cover 3D trigonometry and bearings?+
Yes. Bearings problems (compass directions into triangle equations), 3D problems (finding angles between lines and planes), and applications to vectors and mechanics are all handled step by step with the relevant rule identified.
Is the trigonometry solver free to use?+
SolveGini has a free plan. Guests get 1 solve per day; free accounts get 5 daily solves plus quizzes and flashcards. All step-by-step working is included on the free plan.
Can I use this for A-Level Maths and AP Precalculus?+
Yes. A-Level Pure Maths trig chapters (identities, equations, radian measure, small-angle approximations) and AP Precalculus unit on trig functions and inverse trig are both covered with the standard syllabus language.
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