Trigonometry Calculator

Solve right triangles, compute trig functions, convert angle units, and reference common angle values — all instant, no buttons needed.

Enter any 2 values (sides or angles). The right angle (γ = 90°) is fixed. All results update instantly.

a b c α β γ=90°
0° 90° 180° 270° cos sin
Trigonometric Functions
sin(θ)
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cos(θ)
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tan(θ)
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csc(θ)
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sec(θ)
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cot(θ)
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Inverse Functions
arcsin(value)
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arccos(value)
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arctan(value)
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Degrees ↔ Radians ↔ Gradians
Enter any angle value above
Decimal Degrees ↔ DMS (Degrees Minutes Seconds)
Enter a decimal degree value above

Common Angles Reference

All trig values for standard angles. Values are exact where possible.

Angle (°)sincostancscseccot

FAQ

How do I solve a right triangle?
Enter any two known values (sides or angles) in the Right Triangle Solver. The calculator instantly computes all remaining sides, angles, area, and perimeter using the Pythagorean theorem and basic trig ratios.
What is the difference between degrees, radians, and gradians?
A full circle is 360 degrees, 2π radians, or 400 gradians. Radians are the natural unit in mathematics (arc length = radius × angle). Gradians divide a circle into 400 parts, used in surveying and some European engineering contexts.
What does tan(90°) = ∞ mean?
At 90° (and 270°), the tangent function is undefined because cos(θ) = 0, making the ratio sin/cos approach infinity. The secant is similarly undefined at these angles. The calculator displays ∞ to indicate this.
How do I convert DMS to decimal degrees?
Decimal degrees = Degrees + (Minutes / 60) + (Seconds / 3600). For example, 45° 30' 15" = 45 + 30/60 + 15/3600 = 45.504167°. Use the Conversions tab for instant conversion.
When should I use the law of sines vs. law of cosines?
Use the law of sines (a/sin A = b/sin B = c/sin C) when you know two angles and one side, or two sides and an opposite angle. Use the law of cosines (c² = a² + b² − 2ab cos C) when you know all three sides, or two sides and the included angle. The law of cosines is more general and works for any triangle, not just right triangles.
How do I calculate the area of a triangle using trigonometry?
Use the sine area formula: Area = ½ × a × b × sin(C), where a and b are two known sides and C is the angle between them. For example, a triangle with sides 7 m and 10 m enclosing a 42° angle has an area of ½ × 7 × 10 × sin(42°) ≈ 23.4 m². This works for any triangle where two sides and the included angle are known — no height measurement needed. A related formula, Heron's formula, computes area from all three sides alone.
What are the domains and ranges of trigonometric functions?
Sine and cosine accept any angle and always return values between −1 and 1. Tangent and secant are undefined at 90° and 270° (where cos = 0); cotangent and cosecant are undefined at 0° and 180° (where sin = 0). Inverse functions restrict their domains: arcsin returns −90° to 90°, arccos returns 0° to 180°, and arctan returns −90° to 90°. Knowing these ranges prevents misinterpreting calculator output.
Why are the trig values of 30°, 45°, and 60° special?
These angles produce exact, easily memorized values. sin(30°) = ½, sin(45°) = √2/2, sin(60°) = √3/2. These 'special angles' appear constantly in engineering problems and are worth memorizing. They come from the geometry of equilateral and isosceles right triangles. The calculator's common angles table shows all of them at a glance.

How to Use the Trigonometry Calculator

  1. Enter an angle value — type the angle in degrees or radians. The calculator automatically detects the unit or lets you select it.
  2. View all trig functions — sin, cos, tan, cosec, sec, cot, plus inverse values, all computed instantly.
  3. Solve right triangles — enter two known sides or one side and one angle to find all remaining sides and angles.
  4. Convert angle units — switch between degrees, radians, and gradians with automatic conversion.
  5. Reference the common angles table — for 0°, 30°, 45°, 60°, 90° and their exact trig values for quick lookup.

Understanding Trigonometry Fundamentals

Trigonometry connects angles to ratios of sides in triangles. The three primary functions — sine, cosine, and tangent — are defined for right triangles as follows: sin(θ) = opposite ÷ hypotenuse, cos(θ) = adjacent ÷ hypotenuse, tan(θ) = opposite ÷ adjacent. These ratios extend to the unit circle, where any angle (including those beyond 90°) produces values between −1 and 1 for sin and cos. The reciprocal functions are cosecant (csc = 1/sin), secant (sec = 1/cos), and cotangent (cot = 1/tan).

Engineers use trigonometry constantly. In mechanical engineering, resolving force vectors into components requires sine and cosine. In surveying and construction, the law of sines and cosines determine distances and angles that can't be measured directly. In electrical engineering, AC circuit analysis relies on sinusoidal waveforms where phase angles are calculated using inverse trig functions. Computer graphics and robotics use trigonometric matrices for rotation transformations. Understanding trig fundamentals is essential for any engineering discipline.

Angle units matter — always verify which unit your problem uses. Degrees divide a full circle into 360 parts, intuitive for most people. Radians define an angle as the ratio of arc length to radius, with a full circle equal to 2π radians (≈6.283). Radians are preferred in calculus and higher mathematics because the derivative of sin(x) is cos(x) only when x is in radians. Gradians (or gons) divide a circle into 400 parts, used in some European surveying traditions. The conversion: 180° = π radians = 200 gradians.