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Trigonometry Calculator

Sin, Cos, Tan & all 6 trig functions · Right & oblique triangle solver · Inverse trig · Identity verifier

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All 6 Trig Functions

sin, cos, tan, cot, sec, csc — computed simultaneously

Quick Angle Presets

Worked Examples

Example 1 — Trig Functions at 30°

sin(30°) = 0.5 cos(30°) = √3/2 ≈ 0.8660 tan(30°) = 1/√3 ≈ 0.5774 cot(30°) = √3 ≈ 1.7321 sec(30°) = 2/√3 ≈ 1.1547 csc(30°) = 2

Example 2 — Right Triangle: A=35°, c=10

sin(35°) = a/c a = 10 × sin(35°) = 5.736 cos(35°) = b/c b = 10 × cos(35°) = 8.192 B = 90° − 35° = 55° Area = ½ × 5.736 × 8.192 = 23.49

Example 3 — Oblique SSS: a=5, b=7, c=9

cos(A)=(b²+c²−a²)/(2bc) =(49+81−25)/(126) =0.833 → A=33.56° cos(B)=(a²+c²−b²)/(2ac) =0.583 → B=54.31° C=180°−33.56°−54.31°=92.12° Area=√[s(s−a)(s−b)(s−c)]=17.41

Common Angle Values

Angle Radians sin cos tan
0°0010
30°π/61/2√3/21/√3
45°π/4√2/2√2/21
60°π/3√3/21/2√3
90°π/210Undef.
120°2π/3√3/2−1/2−√3
180°π0−10
270°3π/2−10Undef.
360°2π010

Trigonometry Calculator — Complete Guide

Trigonometry is the branch of mathematics that studies the relationships between the angles and sides of triangles. Derived from the Greek words trigonon (triangle) and metron (measure), trigonometry underpins almost every area of science, engineering, architecture, and navigation. This calculator gives you instant access to all six trigonometric functions, their inverses, right and oblique triangle solving, and identity verification — all in one place.

The Unit Circle and Angle Measurement

The unit circle, a circle of radius 1 centred at the origin, is the foundation of modern trigonometry. Every point on the unit circle corresponds to an angle θ, where the x-coordinate equals cos(θ) and the y-coordinate equals sin(θ). This elegant definition extends trigonometric functions beyond acute angles to all real numbers, enabling the modeling of waves, oscillations, and periodic phenomena.

Angles are measured in either degrees (0° to 360° for a full circle) or radians (0 to 2π for a full circle). Radians are the natural unit because the arc length of a unit circle sector with central angle θ radians is simply θ. To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π.

The Six Trigonometric Functions

For any angle θ in a right triangle with opposite side (a), adjacent side (b), and hypotenuse (c):

  • Sine (sin θ) = opposite / hypotenuse = a/c
  • Cosine (cos θ) = adjacent / hypotenuse = b/c
  • Tangent (tan θ) = opposite / adjacent = sin θ / cos θ
  • Cotangent (cot θ) = adjacent / opposite = cos θ / sin θ = 1/tan θ
  • Secant (sec θ) = hypotenuse / adjacent = 1/cos θ
  • Cosecant (csc θ) = hypotenuse / opposite = 1/sin θ

Note that tan(90°), sec(90°), and cosec(0°) are undefined because they would require division by zero — this calculator detects and displays these cases explicitly.

SOH-CAH-TOA and Right Triangle Solving

SOH-CAH-TOA is the most widely used mnemonic in trigonometry. It reminds students that Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, and Tan = Opposite/Adjacent. Using just two known values from a right triangle — an angle and a side — you can solve for all remaining angles and sides. The Pythagorean theorem (a² + b² = c²) provides an additional constraint when two sides are known.

The right triangle solver in this calculator accepts any valid combination of 2 known values and automatically determines which formula chain to use, showing every step of the working.

Law of Sines and Law of Cosines

For oblique (non-right) triangles, two powerful laws apply. The Law of Sines states that a/sin(A) = b/sin(B) = c/sin(C), where lowercase letters are sides and uppercase are opposite angles. It is ideal for ASA, AAS, and SSA cases. The SSA case is special — it is the "ambiguous case" where two distinct triangles may exist with the same data.

The Law of Cosines generalises the Pythagorean theorem: c² = a² + b² − 2ab·cos(C). It handles SSS (all three sides known) and SAS (two sides and the included angle) cases. For SSS, the area is calculated using Heron's formula: Area = √[s(s−a)(s−b)(s−c)] where s = (a+b+c)/2.

Inverse Trigonometric Functions

Inverse trig functions answer the question "what angle produces this value?" arcsin(x) gives the angle whose sine is x, arccos(x) gives the angle whose cosine is x, and arctan(x) gives the angle whose tangent is x. The domain of arcsin and arccos is restricted to [−1, 1], while arctan accepts all real numbers. The outputs are expressed in both degrees and radians.

Real-World Applications of Trigonometry

Trigonometry has countless practical applications. In architecture and engineering, it is used to calculate roof slopes, bridge stresses, and structural loads. In navigation, sailors and pilots use the law of sines and cosines to determine position and bearing. GPS systems rely on trigonometric calculations to triangulate location. In physics, wave mechanics, optics, and acoustics all depend on sinusoidal functions. In computer graphics, every rotation, transformation, and 3D projection uses a trigonometric matrix.

Signal processing converts waveforms into frequency components using the Fourier transform, which decomposes any periodic signal into sums of sines and cosines. Medical imaging technologies like MRI and CT scans use similar principles. Even music theory, with its harmonics and overtones, is rooted in the mathematics of trigonometric waves.

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Frequently Asked Questions

What is SOH-CAH-TOA?
SOH-CAH-TOA is a mnemonic for the three primary trigonometric ratios in a right triangle. SOH: Sine = Opposite / Hypotenuse. CAH: Cosine = Adjacent / Hypotenuse. TOA: Tangent = Opposite / Adjacent. This memory aid helps students recall which sides to use when computing sin, cos, and tan. For example, if angle A = 35° and the hypotenuse is 10, then the opposite side (a) = 10 × sin(35°) ≈ 5.74.
What is the difference between degrees and radians?
Degrees and radians are two units for measuring angles. A full rotation is 360° or 2π radians. To convert from degrees to radians: multiply by π/180. To convert from radians to degrees: multiply by 180/π. Key values: 90° = π/2, 180° = π, 270° = 3π/2, 360° = 2π. Radians are preferred in mathematics and physics because they make many formulas simpler — for instance, the arc length formula s = rθ only works when θ is in radians.
How do I solve a right triangle?
To solve a right triangle (find all 3 sides and 3 angles) you need at least 2 known values, including at least one side length. The key relationships are: sin(A) = a/c, cos(A) = b/c, tan(A) = a/b, and the Pythagorean theorem a² + b² = c². The three angles always sum to 180°, with one angle fixed at 90°. This calculator automatically detects your input combination (angle + side, two sides, etc.) and applies the correct formula with step-by-step working shown.
What are the 6 trigonometric functions?
The six trig functions are: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent, cot(θ) = adjacent/opposite = 1/tan(θ), sec(θ) = hypotenuse/adjacent = 1/cos(θ), and csc(θ) (cosecant) = hypotenuse/opposite = 1/sin(θ). Cot, sec, and csc are the reciprocals of tan, cos, and sin respectively. Tan and sec are undefined when cos(θ) = 0 (at 90°, 270°). Cot and csc are undefined when sin(θ) = 0 (at 0°, 180°, 360°).
What is the law of sines used for?
The law of sines (a/sin A = b/sin B = c/sin C) is used to solve oblique (non-right) triangles in the ASA, AAS, and SSA configurations. In the ASA case you know two angles and the included side; in AAS you know two angles and a non-included side. The SSA case is the ambiguous case — given two sides and a non-included angle, there may be 0, 1, or 2 valid triangles. This calculator detects and flags the ambiguous case and shows the primary acute solution.
What are Pythagorean identities?
The three Pythagorean trig identities are: (1) sin²θ + cos²θ = 1 — the most fundamental identity, derived directly from the Pythagorean theorem on the unit circle; (2) 1 + tan²θ = sec²θ — obtained by dividing identity (1) by cos²θ; (3) 1 + cot²θ = csc²θ — obtained by dividing identity (1) by sin²θ. These identities hold for every angle and are used extensively to simplify trigonometric expressions and solve equations.
How do I convert degrees to radians?
To convert degrees to radians, multiply by π/180 (approximately 0.017453). Common conversions: 30° = π/6 ≈ 0.5236 rad, 45° = π/4 ≈ 0.7854 rad, 60° = π/3 ≈ 1.0472 rad, 90° = π/2 ≈ 1.5708 rad, 180° = π ≈ 3.1416 rad, 270° = 3π/2 ≈ 4.7124 rad, 360° = 2π ≈ 6.2832 rad. To reverse (radians to degrees), multiply by 180/π. This calculator supports both units — use the DEG/RAD toggle to switch at any time.