Trigonometry Calculator

A trigonometry calculator computes the six trigonometric functions (sine, cosine, tangent, and their reciprocals cosecant, secant, and cotangent) for any angle, and solves triangles given sides and angles. It is used by students in GCSE and A-level mathematics, engineers, physicists, and surveyors working with angles and periodic functions.

How to Use the Trigonometry Calculator

1. Select your mode: evaluate a trig function, solve a right triangle, or solve an oblique triangle.
2. Enter the angle in degrees or radians, then select your preferred unit.
3. For triangle solving, enter the known sides and angles (at least one side plus one other value).
4. Click Calculate to see all six trig values or the full triangle solution.
5. Review the unit circle diagram and exact values displayed alongside the decimal results.

The Formula

For a right-angled triangle with angle theta, opposite side O, adjacent side A, and hypotenuse H:

sin(theta) = O / H cos(theta) = A / H tan(theta) = O / A = sin(theta) / cos(theta) cosec(theta) = 1 / sin(theta) sec(theta) = 1 / cos(theta) cot(theta) = 1 / tan(theta)

For oblique triangles, the Sine Rule and Cosine Rule apply: Sine Rule: a / sin(A) = b / sin(B) = c / sin(C) Cosine Rule: a^2 = b^2 + c^2 - 2bc x cos(A)

Real-World Example

A ladder leans against a wall. The ladder is 5 metres long and makes an angle of 65 degrees with the ground. How high up the wall does it reach?

The height is the opposite side, the ladder is the hypotenuse.

Height = H x sin(theta) = 5 x sin(65°) = 5 x 0.9063 = 4.53 metres.

The distance from the wall along the ground = 5 x cos(65°) = 5 x 0.4226 = 2.11 metres.

Check using Pythagoras: 4.53^2 + 2.11^2 = 20.52 + 4.45 = 24.97 ≈ 25 = 5^2. Correct.

Trigonometry in Science and Engineering

Trigonometric functions appear throughout science and engineering wherever waves, oscillations, or angles are involved. In physics, simple harmonic motion (springs, pendulums) is described by sine and cosine functions. Sound and light are modelled as sine waves, and Fourier analysis decomposes any complex wave into a sum of trigonometric functions. In engineering, surveyors use trigonometry to measure distances and heights that are inaccessible directly. Architects and structural engineers apply it to calculate forces in trusses and roof structures. In navigation, the haversine formula uses trigonometry to calculate great-circle distances between two points on a sphere. Signal processing, computer graphics, robotics, and GPS all rely heavily on trigonometric mathematics.

Frequently Asked Questions

What is the difference between degrees and radians? Degrees and radians are two ways of measuring angles. A full circle is 360 degrees or 2*pi radians. Radians are the natural unit in mathematics because they make calculus formulas simpler. To convert from degrees to radians, multiply by pi/180. To convert from radians to degrees, multiply by 180/pi. Most scientific calculators have a mode switch (DEG/RAD) to control which unit is in use.

What are the exact values of common angles? For 30°: sin = 1/2, cos = sqrt(3)/2, tan = 1/sqrt(3). For 45°: sin = cos = 1/sqrt(2), tan = 1. For 60°: sin = sqrt(3)/2, cos = 1/2, tan = sqrt(3). These exact values come from equilateral and isosceles right triangles and are required knowledge for GCSE and A-level exams.

When do I use the Sine Rule versus the Cosine Rule? Use the Sine Rule when you know two angles and one side (AAS or ASA), or two sides and a non-included angle (SSA). Use the Cosine Rule when you know two sides and the included angle (SAS) or all three sides (SSS). The Cosine Rule is also more numerically stable for triangles where angles are close to 0 or 180 degrees.

What does it mean when tan is undefined? Tangent is undefined at 90° and 270° (or pi/2 and 3pi/2 radians) because cosine equals zero at these angles and tan = sin/cos. At these points, the tangent function shoots off to positive or negative infinity. Geometrically, this corresponds to a right triangle where the opposite side would be infinitely long relative to the adjacent side.

Solving an oblique triangle step by step

The calculator returns function values for an angle. It does not solve a triangle for you, so the working below is the part a reader does with those values. Take a triangle with two sides and the included angle: a = 7.0 cm, b = 9.0 cm, and angle C = 52 degrees between them.

Start with the cosine rule to find the third side. The rule gives c^2 = a^2 + b^2 - 2ab cos(C), so c^2 = 49 + 81 - 126 x 0.6156615 = 130 - 77.57335 = 52.42665, and c = 7.24063 cm.

Then use the sine rule for the two remaining angles. The rule is a / sin(A) = c / sin(C), rearranged as sin(A) = a sin(C) / c. That gives sin(A) = 7 x 0.7880108 / 7.24063 = 0.761823, so A = 49.6252 degrees. Angle B follows from the angle sum: B = 180 - 52 - 49.6252 = 78.3748 degrees.

The check matters because the sine rule cannot tell the difference between an angle and its supplement. Both checks here return the same B, which is a good sign. A triangle with two possible shapes shows up as a second valid value, and the section below covers that case.

The case with two answers

Give the same method a triangle with a = 7.0 cm, b = 9.0 cm and angle A = 40 degrees, and the sine rule returns sin(B) = 9 x sin(40) / 7 = 0.826441. Two angles between 0 and 180 degrees have that sine: 55.7349 degrees and 124.2651 degrees. Both are legitimate triangles. The first leaves C = 84.2651 degrees and a third side of 10.84 cm, and the second leaves C = 15.7349 degrees and a third side of 2.95 cm.

Solve the third side before deciding. If both candidate triangles give a positive side length consistent with the given sides, the data does not determine one triangle, and the problem needs more information. This is the only configuration in triangle solving where the answer is not unique: three sides, two angles with a side, or two sides with the included angle each fix a single triangle.

Exact values at the standard angles

The results below are the values the calculator displays, written to six decimal places.

The exact forms behind those decimals are worth knowing: sin 30 = 1/2, cos 30 = sqrt(3)/2 = 0.866025, tan 30 = 1/sqrt(3) = 0.577350, sin 45 = cos 45 = 1/sqrt(2) = 0.707107, sin 60 = sqrt(3)/2, and tan 60 = sqrt(3) = 1.732051.

Degrees, radians and gradians

One full turn is 360 degrees, 2 pi radians, or 400 gradians. The calculator's unit selector offers all three, and the conversion runs through radians internally.

• 180 degrees is pi radians, which is 3.141593 to six decimals.
• One degree is pi / 180 radians, which is 0.01745329.
• One radian is 180 / pi degrees, which is 57.295780.

Radians are the unit to reach for whenever the answer feeds a formula rather than a diagram. The derivative of the sine function is the cosine only when the angle is measured in radians, and the same applies to the small-angle approximation sin(x) is approximately x. Degrees are easier for reading a protractor and for most survey work, which is why the selector sits next to the input rather than in a settings menu.

Where the answers can surprise you

Four behaviours are worth knowing before you trust a displayed number.

• The inverse mode returns principal values. arcsin and arctan answer between minus 90 and 90 degrees, and arccos answers between 0 and 180 degrees. When the geometry admits a second angle, the calculator gives the principal one and the working above is how you find the other.
• The inverse mode covers arcsin, arccos and arctan. The reciprocal functions are not in that menu, so cot, sec and cosec are read from the forward mode results or by taking a reciprocal by hand.
• Tangent at 90 degrees prints a very large number rather than an error. The radian value of 90 degrees cannot be stored exactly in binary, so the cosine is a tiny non-zero quantity and the division produces roughly 1.6331239e+16. Treat any tangent above about 1e15 as an undefined case.
• The displayed values carry about eight significant figures. Subtracting two nearly equal values, such as cos 0 minus cos 0.000001, loses precision, and the loss belongs to the arithmetic rather than to the calculator.

The definitions of the trigonometric functions used here follow ISO 80000-2, Quantities and units, Part 2: Mathematics. The radian is defined as a dimensionless derived unit in the BIPM's SI Brochure, ninth edition (2019), which also lists the degree as a unit accepted for use with the SI.

Also try these free tools:

• Triangle Calculator
• Pythagorean Theorem Calculator
• Scientific Calculator