Trigonometry Basics
8 min read · Updated September 26, 2026 · by the SolveCalcPro editorial team
Trigonometry connects the angles of a triangle to the ratios of its sides. Surveyors, pilots, engineers and game designers all use it to calculate distances they cannot measure directly. Start with right triangles and three ratios.
SOH-CAH-TOA
Label the sides relative to an angle θ: the hypotenuse is the longest side, the opposite side faces θ and the adjacent side touches it. Then sin θ = opposite ÷ hypotenuse, cos θ = adjacent ÷ hypotenuse and tan θ = opposite ÷ adjacent.
Finding an angle
In a 3-4-5 right triangle, the angle opposite the side of length 3 has sin A = 3/5 = 0.6. Use the inverse function: A = sin⁻¹(0.6) ≈ 36.87°. The trigonometry calculator does this in one step.
Finding a side: the height of a tree
You stand 20 m from a tree and measure an angle of elevation of 35° to its top. tan 35° = height ÷ 20, so height = 20 × tan 35° ≈ 20 × 0.7002 ≈ 14.0 m (plus your eye height).
Special angles
| θ | sin | cos | tan |
|---|---|---|---|
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
Beyond right triangles
The law of sines (a ÷ sin A = b ÷ sin B) and the law of cosines (c² = a² + b² − 2ab cos C) handle any triangle. The triangle calculator uses them.
Check your calculator mode
Make sure your calculator is in degrees when angles are in degrees. See degrees vs radians, and try the trigonometry quiz.
Practice questions
Test yourself, then tap to check. Answers are calculated by the tools linked below.
- Find sin, cos and tan of 45°.
Show answer
sin: 0.7071067812 · cos: 0.7071067812 · tan: 1
- Solve a triangle with sides 5, 12 and 13.
Show answer
Side a: 5 · Side b: 12 · Side c: 13
- A right triangle has legs 9 and 12. Find the hypotenuse.
Show answer
Hypotenuse c: 15
Frequently asked questions
What does SOH-CAH-TOA mean?
Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
How do I find an angle from two sides?
Pick the ratio that uses those two sides, then use the inverse trig function.
What is an angle of elevation?
The angle above the horizontal when you look up at something.
Does trigonometry only work for right triangles?
The basic ratios do. The laws of sines and cosines extend it to all triangles.
Try it yourself
Trigonometry CalculatorFind sin, cos, tan, csc, sec and cot of any angle, or the inverse trig functions.Triangle CalculatorSolve any triangle from three sides, two sides and an angle, or two angles and a side.Pythagorean Theorem CalculatorFind the hypotenuse or a missing leg of a right triangle, plus area, perimeter and angles.Keep learning
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