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📐 Trigonometry Calculator

Evaluate sin, cos, tan, and inverse trig functions; convert degrees and radians; solve right-triangle ratios.

sin, cos, tan and Inverses — Degrees or Radians

BrainyCalculators editorial insight — unique to this tool

SOH-CAH-TOA maps sides in right triangles — 30°-60°-90° shortcuts (1, √3, 2) speed ACT/SAT trig. Surveying and navigation use bearing angles with sin/cos components. Inverse trig returns angle from ratio.

When to use this calculator

Use for trig function evaluation in triangles. For full expression calculator with many functions, use Scientific.

Known side lengths on a triangle?

This page evaluates trig functions on angles. For SSS/SAS triangle solving and area, use the Triangle Calculator →

Trig Functions — Enter an Angle

Inverse Trig — Enter a Value

What is a Trigonometry Calculator?

A trigonometry calculator evaluates trig functions on angles and solves right-triangle ratio problems. It handles degree/radian mode and inverse functions where defined.

Use this page when the inputs are angles or trig ratios. For full triangle solving with three sides or SAS, the Triangle Calculator is more specific.

For general keypad arithmetic including logs and powers, use the Scientific Calculator instead.

SOH-CAH-TOA Explained

SOH
sin θ = Opp / Hyp
CAH
cos θ = Adj / Hyp
TOA
tan θ = Opp / Adj

Unit Circle Key Angles

Degrees Radians sin cos tan
0° 0 0 1 0
30° π/6 ½ √3/2 1/√3
45° π/4 √2/2 √2/2 1
60° π/3 √3/2 ½ √3
90° π/2 1 0 undef
180° π 0 −1 0
270° 3π/2 −1 0 undef
360° 2π 0 1 0

How the Trigonometry Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

  1. Read the values and operation selected.
  2. Normalize signs, decimals, fractions, or units if needed.
  3. Apply the mathematical rule or formula.
  4. Format the answer and any intermediate values for checking.

Assumptions and Limits

  • Inputs must be within the supported domain of the operation.
  • Decimal answers may be rounded for readability.
  • Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

In a right triangle, sine (sin) is the ratio of the opposite side to the hypotenuse, cosine (cos) is adjacent over hypotenuse, and tangent (tan) is opposite over adjacent. These ratios depend only on the angle, not the triangle size.

SOH-CAH-TOA is a mnemonic: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent. It is the most common way to remember the three primary trig functions.

Both measure angles. A full circle is 360° or 2π radians. To convert degrees to radians multiply by π/180. To convert radians to degrees multiply by 180/π. Radians are the natural unit used in calculus and most scientific contexts.

The unit circle is a circle with radius 1 centred at the origin. For any angle θ, the point on the unit circle is (cos θ, sin θ). It provides a visual way to understand trig values for all angles, including those greater than 90°.

tan θ = sin θ / cos θ. At 90°, cos θ = 0, and division by zero is undefined. Similarly, cot(0°) = cos(0°)/sin(0°) = 1/0, which is undefined. These are called singularities of the trig functions.

Real-World Applications

🏗️
Architecture & Construction
Architects and structural engineers use trigonometry to calculate roof pitch angles, rafter lengths, wall tie spacings, and the horizontal and vertical components of inclined structural forces. A roof with a 35° pitch and a 6-metre span requires rafters of length 6 / (2 × cos 35°) = 3.66m — a direct application of the cosine ratio. Staircase geometry, the angle and run of access ramps, and the setting-out of curved structures all involve trigonometric calculations.
🔌
AC Electrical Engineering
Alternating current analysis relies heavily on trigonometry. Voltage and current in AC circuits are sinusoidal: V(t) = Vmax × sin(ωt + φ). Impedance in RLC circuits is described by an impedance triangle where the phase angle between voltage and current is arctan(X/R). Power factor — the cosine of the impedance phase angle — determines the fraction of apparent power (volt-amperes) converted to real work (watts). Every AC power system calculation involves trigonometric functions.
🔭
Physics — Wave Motion & Optics
Trigonometric functions describe all wave phenomena: sound, light, electromagnetic radiation, water waves, and quantum wavefunctions. Snell's law of refraction (n₁ sin θ₁ = n₂ sin θ₂) governs how light bends at the interface between media — determining the design of lenses, prisms, and optical fibres. The diffraction grating equation (d sin θ = mλ) uses the sine function to relate wavelength to diffraction angle, underpinning spectroscopy and the analysis of atomic structure.
🎵
Audio Engineering & Signal Processing
Audio signals are decomposed into sinusoidal components using Fourier analysis — every complex waveform (speech, music, noise) is mathematically represented as a sum of sine waves of different frequencies, amplitudes, and phases. Equalisation, compression, reverb, and all digital audio effects operate on these trigonometric components. The discrete Fourier transform (DFT) and its fast implementation (FFT) use trigonometric sum formulas to convert between time-domain waveforms and frequency-domain spectra.
🕹️
3D Game Development & Animation
Game engines apply rotation transformations to 3D objects using rotation matrices built from cosine and sine values of rotation angles. Rotating an object 45° around the Y-axis involves computing cos(45°) and sin(45°) and applying them to a 4×4 transformation matrix. Camera perspective projection, character movement towards a target bearing, projectile trajectory calculation, and inverse kinematics for animated character limbs all use trigonometric functions extensively.
🌐
GPS & Geodesy
GPS receivers calculate position by measuring signal travel times from multiple satellites and solving the system of spherical trigonometry equations that determines the intersection point on Earth's curved surface. The haversine formula — a spherical trigonometry formula involving arcsin, cos, and sin — computes great circle distance between two latitude/longitude coordinates on Earth's surface. Every mapping application, navigation system, and location-based service is built on these spherical trigonometric foundations.

Common Mistakes

1
Using degrees when the calculator expects radians (and vice versa)
The most common trigonometry error is supplying an angle in degrees to a function that expects radians. sin(90°) = 1.0, but sin(90 radians) ≈ 0.894 — a completely different result. Most scientific calculators display a DEG/RAD mode indicator; programming languages (Python's math.sin, JavaScript's Math.sin) default to radians. Convert degrees to radians by multiplying by π/180 (≈ 0.01745); convert radians to degrees by multiplying by 180/π. Always verify angle mode before computing.
2
Forgetting that inverse trig functions return values in a limited range
arcsin returns values in [−90°, 90°]; arccos returns values in [0°, 180°]; arctan returns values in [−90°, 90°]. These ranges mean that arcsin(0.5) = 30° — but 150° also has sin(150°) = 0.5. When solving for angles in contexts where the answer might lie outside the principal range (e.g. a triangle with an obtuse angle, or a direction in the third or fourth quadrant), the inverse trig result must be adjusted using the known geometric context. The atan2(y, x) function handles the full 360° range correctly for direction angles.
3
Misidentifying opposite, adjacent, and hypotenuse sides in non-standard orientations
SOH-CAH-TOA labels are defined relative to the angle in question, not the triangle's orientation. "Opposite" is the side across from the angle being used; "adjacent" is the side next to it (that is not the hypotenuse). When a right triangle is drawn with the right angle at a non-standard corner or rotated, students often misidentify which side is opposite and which is adjacent relative to the reference angle. Draw the triangle, label the right angle, identify the reference angle, and then label opposite and adjacent explicitly before applying any trig ratio.
4
Using a right-triangle formula on a non-right triangle
SOH-CAH-TOA applies only in right triangles. For non-right (oblique) triangles, the Law of Sines or Law of Cosines must be used. A common error is measuring an angle in an oblique triangle, then applying sin(angle) = opposite/hypotenuse when there is no hypotenuse in the triangle (hypotenuse is defined only in right triangles as the side opposite the right angle). If none of the three angles is 90°, the triangle is oblique and requires the general laws.
5
Rounding intermediate results before the final calculation
Trigonometric calculations are sensitive to intermediate rounding because small angle differences produce disproportionate changes in trig function values near 0° and 90°. Rounding sin(89.5°) to sin(90°) = 1.0 introduces a 1% error; rounding an angle at an earlier step can cascade through multiple calculation steps, amplifying the error. Always carry full calculator precision through all intermediate steps and round only the final answer to the required number of significant figures.

Standard Angle Trigonometric Values Quick Reference

Angle Radians sin cos tan
0° 0 0 1 0
30° π/6 0.500 0.866 0.577
45° π/4 0.707 0.707 1.000
60° π/3 0.866 0.500 1.732
90° π/2 1.000 0 undefined

References

  1. Larson, R. Trigonometry, 10th edition. Cengage Learning, 2016.
  2. Stewart, J., Redlin, L. and Watson, S. Precalculus: Mathematics for Calculus. Cengage, 2015.
  3. Weisstein, E.W. Trigonometric Functions. MathWorld — Wolfram Research, 2024.
  4. NIST. Digital Library of Mathematical Functions — Trigonometric Functions. dlmf.nist.gov, 2024.
  5. Euclid. Elements. c. 300 BCE.

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