AI Trigonometry Solver
Trigonometry links the angles of a triangle to its side lengths. This AI trigonometry solver evaluates sine, cosine and tangent, works with degrees or radians and explains which relationship is being used.
SOH CAH TOA in one minute
In a right-angled triangle, the three main ratios compare sides to an angle θ:
- sin θ = opposite ÷ hypotenuse
- cos θ = adjacent ÷ hypotenuse
- tan θ = opposite ÷ adjacent
The hypotenuse is the longest side, opposite the right angle. “Opposite” and “adjacent” depend on which angle you are using, so label your triangle before you calculate.
Exact values to memorise
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° (π/6) | 1/2 | √3/2 | 1/√3 |
| 45° (π/4) | √2/2 | √2/2 | 1 |
| 60° (π/3) | √3/2 | 1/2 | √3 |
| 90° (π/2) | 1 | 0 | undefined |
The identity sin²θ + cos²θ = 1 holds for every angle and is the basis for most simplification questions.
Degrees versus radians
Degrees divide a full turn into 360 parts; radians measure the angle by arc length, so a full turn is 2π. To convert, multiply degrees by π/180, or radians by 180/π. Check your calculator mode before every trig question, because a wrong mode is the most common reason an answer is off. Type a degree sign (for example sin(30°)) so the solver knows you mean degrees. Graphs of these functions can be explored in the graphing solver, and rates of change are in the calculus solver.
Worked examples
Example 1: Evaluate sin(30°) + cos(60°)
- sin 30° = 1/2.
- cos 60° = 1/2.
- Add: 1/2 + 1/2 = 1.
Final answer: 1
Example 2: A right triangle has opposite side 5 and hypotenuse 13. Find sin θ, cos θ and tan θ.
- sin θ = 5/13.
- Find the adjacent side with Pythagoras: √(13² − 5²) = √144 = 12.
- cos θ = 12/13 and tan θ = 5/12.
Final answer: sin θ = 5/13, cos θ = 12/13, tan θ = 5/12
Example 3: Convert π/6 radians to degrees and find its sine
- Multiply by 180/π: (π/6)(180/π) = 30°.
- sin 30° = 1/2.
Final answer: 30°; sin = 1/2
Example 4: Solve sin x = 1/2 for 0° ≤ x < 360°
- The calculator gives the principal value x = 30°.
- Sine is also positive in the second quadrant: x = 180° − 30° = 150°.
Final answer: x = 30° or x = 150°
Common mistakes to avoid
- Using the wrong calculator mode (degrees versus radians).
- Labelling opposite and adjacent from the wrong angle.
- Giving only the first solution of a trig equation and missing the second quadrant.
- Writing sin²θ as sin(θ²); it means (sin θ)².
- Forgetting that tan 90° is undefined.
Frequently asked questions
How do I enter degrees?
Add the degree symbol after the number, for example sin(30°). The input keypad includes it.
What is the unit circle?
A circle of radius 1 centred on the origin. The point at angle θ has coordinates (cos θ, sin θ), which is why sine and cosine lie between −1 and 1.
Can the solver find missing sides of triangles?
Yes. Describe the triangle in words or numbers in AI mode and it will choose between the trig ratios, Pythagoras and the sine and cosine rules.
Why does tan 90° have no value?
Because cos 90° = 0 and tan θ = sin θ ÷ cos θ, so you would be dividing by zero.
Related math tools and guides
Explore all math tools
Math solvers
Learning tools
How-to guides
AI-generated solutions are for learning purposes. Always verify important answers.