Trigonometry Calculator — Sin, Cos, Tan & Right Triangles

What Is a Trigonometry Calculator?

A trigonometry calculator answers the three questions students ask most: what are sin, cos and tan of this angle? What is sin, cos or tan of this triangle? And what angle gives this ratio? Give it an angle and it returns the ratios — sin 30° = 0.5, tan 45° = 1. Give it a triangle's sides and it returns the ratios and the angle — opposite 3, hypotenuse 5 gives sin θ = 3/5 = 0.6 and θ = 36.87°. Give it a ratio and inverse trig returns the angle — arcsin(0.5) = 30°. Every answer comes with the step-by-step working so you can verify it by hand — including the famous trap: if sin θ = 3/5, tan θ is 3/4, not 4/5.

How to Use the Trigonometry Calculator

The tool has four modes so you can start with whatever you already know.

1. Choose a mode that matches your problem (or tap a preset to load a worked example).
2. Enter your numbers — angles in degrees, sides in any unit.
3. Select Calculate to generate the result cards.
4. Read the note under the cards for the step-by-step working and a check you can verify by hand.

The built-in check re-derives the answer a second way — the hypotenuse from Pythagoras against the ratio, or sin²θ + cos²θ = 1 — so a wrong keystroke shows up immediately.

What Is Trigonometry?

Trigonometry is the study of the relationships between the angles and sides of triangles — the word comes from the Greek for "triangle measurement". Its three core functions, sine (sin), cosine (cos) and tangent (tan), are ratios of a right triangle's sides, and they are the same for every triangle that shares the same angle. That is the whole power of the subject: measure the angle once, and you know the ratios for every triangle with that angle.

SOH-CAH-TOA — The One Memory Trick

SOH-CAH-TOA packs all three definitions into one word: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. Say it out loud while you label the triangle — opposite, adjacent, hypotenuse — and the formula writes itself. For the 3-4-5 triangle with θ opposite the side 3: sin θ = 3/5 = 0.6, cos θ = 4/5 = 0.8, tan θ = 3/4 = 0.75.

The Special Angles: 0°, 30°, 45°, 60°, 90°

Five angles appear in almost every textbook problem, and their sin, cos and tan values are worth knowing by heart. They all come from two triangles: the 30-60-90 triangle (half an equilateral triangle) and the 45-45-90 triangle (half a square).

Inverse Trig — Finding the Angle From a Ratio

Inverse trig runs the functions backwards: arcsin, arccos and arctan take a ratio and return the angle. If sin θ = 0.5, then θ = arcsin(0.5) = 30°. If tan θ = 1, then θ = arctan(1) = 45°. If cos θ = 0.5, then θ = arccos(0.5) = 60°. The catch is the range: arcsin and arccos only accept ratios between −1 and 1 (a ratio of sides can never be bigger than 1 for sin and cos), while arctan accepts any number. Enter arcsin(2) and the calculator rejects it — no angle has a sine of 2.

Trigonometry Calculator Worked Examples

Every row below is computed exactly — hand-verified in Python — so you can check each one by hand.

The 3-4-5 Trap: Why tan θ Is 3/4, Not 4/5

The trap is reading sin θ = 3/5 and guessing tan θ = 4/5 — swapping the numbers without thinking. Tangent is opposite over adjacent, not opposite over hypotenuse. With opposite 3 and hypotenuse 5, Pythagoras finds the missing side first: adjacent² = 5² − 3² = 25 − 9 = 16, so adjacent = 4. Then tan θ = 3/4 = 0.75. The check: sin²θ + cos²θ = 0.6² + 0.8² = 0.36 + 0.64 = 1 — the identity holds, so the sides are right.

The Ladder Problem: A Real-World Right Triangle

A 10-foot ladder leans against a wall at 70° to the ground. How high does it reach? The ladder is the hypotenuse (10), the height is the opposite side: height = 10 × sin 70° = 10 × 0.9397 = 9.3969 ft. The foot of the ladder sits 10 × cos 70° = 10 × 0.3420 = 3.4202 ft from the wall. Check: 9.3969² + 3.4202² = 88.30 + 11.70 = 100 = 10² — Pythagoras confirms it.

How to Find Sin, Cos and Tan Step by Step

Follow this recipe for the classic example — sin θ = 3/5, find tan θ.

1. Label the triangle: Opposite the angle θ is 3, the hypotenuse is 5. The third side is the adjacent side — unknown yet.
2. Find the adjacent side with Pythagoras: adjacent² = hypotenuse² − opposite² = 5² − 3² = 25 − 9 = 16, so adjacent = 4.
3. Write sin θ: sin θ = opposite/hypotenuse = 3/5 = 0.6.
4. Write cos θ: cos θ = adjacent/hypotenuse = 4/5 = 0.8.
5. Write tan θ: tan θ = opposite/adjacent = 3/4 = 0.75 — not 4/5.
6. Check with the identity: sin²θ + cos²θ = 0.6² + 0.8² = 0.36 + 0.64 = 1 ✓. And find the angle if you need it: θ = arcsin(0.6) = 36.87°.

Common Mistakes With Trigonometry

1. Saying tan θ = 4/5 when sin θ = 3/5 — the #1 trap

Tangent is opposite over adjacent, not opposite over hypotenuse. With opposite 3 and hypotenuse 5, the adjacent side is 4 (Pythagoras: 5² − 3² = 16), so tan θ = 3/4 = 0.75. The number 4/5 is cos θ — mixing up tan and cos is exactly the trap.

2. Labelling opposite and adjacent from the wrong angle

Opposite and adjacent are defined relative to the angle θ you are working with. In a 3-4-5 triangle, if θ sits opposite the side 3, then opposite = 3 and adjacent = 4. If you switch to the other acute angle (53.13°), opposite and adjacent swap: sin becomes 4/5 and cos becomes 3/5. Always label the triangle for the angle in your question first.

3. Using the hypotenuse as the adjacent side

The adjacent side is the leg next to θ — the hypotenuse is never the adjacent side and never the opposite side. In tan θ = opposite/adjacent, neither number is ever the hypotenuse. If your tan comes out using 5 in a 3-4-5 triangle, you have used the wrong side.

4. Degrees vs radians

Most calculators and code default to radians. sin(30°) = 0.5, but sin(30 radians) = −0.9880. If your answer looks nothing like the table above, check the angle mode first — this calculator works in degrees throughout.

5. Taking arcsin of a number bigger than 1

sin θ and cos θ can never be bigger than 1, so arcsin(2) and arccos(1.5) have no answer. If your ratio comes out above 1, the sides are mislabelled — usually the hypotenuse was entered as the opposite side. Arctan has no such limit: any number works.

6. Forgetting tan 90° is undefined

tan 90° = sin 90°/cos 90° = 1/0 — division by zero, so it is undefined, not 0 and not 1. The same goes for any angle where cos θ = 0. If your calculator shows an enormous number (like 1.6 × 10¹⁶), that is its way of saying "undefined".

Benefits and Limitations

Benefits

• Finds sin, cos and tan from an angle, from sides, or from a ratio — all in one tool.
• Inverse trig (arcsin, arccos, arctan) returns the angle in degrees, with range checks that reject impossible ratios.
• The right-triangle solver finds every missing side and angle from the hypotenuse plus one angle or one side.
• Every answer carries the step-by-step working — SOH-CAH-TOA, Pythagoras, the identity check — so homework can be checked by hand.
• The special-angle table (0°, 30°, 45°, 60°, 90°) is explained, not just listed.
• Responsive on phones, tablets and desktops.

Limitations

• Right triangles only for the solver — non-right triangles need the law of sines and cosines (see the Triangle Calculator).
• Angles are in degrees only; radians are not accepted.
• Results use JavaScript floating-point arithmetic, so ratios and angles are rounded to 4 decimal places.

Quick Definitions

Related Math Tools

Trigonometry Calculator Methodology

This calculator derives every answer from the standard trigonometry definitions, using JavaScript floating-point arithmetic (ratios and angles rounded to 4 decimal places).

Trig-from-angle mode evaluates sin, cos and tan directly from the angle in degrees (tan 90° is reported as undefined, since cos 90° = 0). Trig-from-sides mode finds the hypotenuse with Pythagoras from the two legs, then all three ratios and the angle with arctan(opposite/adjacent). Inverse-trig mode rejects sin/cos ratios outside [−1, 1] before computing, since no angle has a sine or cosine beyond that range. The right-triangle solver takes the hypotenuse plus one angle or one side, finds the missing sides with sin/cos or Pythagoras, and the other acute angle as 90° minus the known angle. Every result carries the identity check sin²θ + cos²θ = 1 and a Pythagoras cross-check of the three sides. Inputs must be positive sides and angles strictly between 0° and 90° in the solver; anything else is rejected before any calculation. The page is intended as an educational calculator and verification tool.

Mathematical References

• Khan Academy – Intro to Trigonometric Ratios
• Math is Fun – Sine, Cosine, Tangent
• Math is Fun – Inverse Sine, Cosine, Tangent
• Khan Academy – Solving for a Side with the Law of Sines
• Wikipedia – Trigonometric Functions