The Trigonometry Calculator computes the values of sin, cos, tan, csc, sec, and cot for a given angle.
Supports degree and radian input, inverse functions (arcsin, arccos, arctan), a special angles table with exact values, 19 trigonometric identities, and a unit circle SVG visualization.
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Calculator information
📋 How to use this calculator
- Select input unit: degrees (0-360°) or radians (0-2π).
- Enter the angle value, for example: 30, 45, 90, or pi/4 (the calculator parses pi automatically).
- Select a function: sin, cos, tan, csc, sec, cot, or inverse (arcsin, arccos, arctan).
- Press Calculate for the numeric result plus exact values for special angles (30°, 45°, 60°, etc.).
- View the SVG unit-circle visualization showing the angle position, coordinates (cos θ, sin θ), and quadrant.
- Check the table of 19 trigonometric identities for Pythagorean, double-angle, sum-angle, and half-angle formulas.
- Tip: For quick conversion, 1 rad = 57.2958° or π rad = 180°; use degrees for geometry/surveying, radians for calculus/physics.
🧮 Trigonometric Functions & Pythagorean Identities
sin²θ + cos²θ = 1 ; tan θ = sin θ / cos θ ; θ(rad) = θ(°) × π/180
- Unit circle: x = cos θ, y = sin θ for a point at radius 1
- csc θ = 1/sin θ, sec θ = 1/cos θ, cot θ = 1/tan θ
- Special angles: sin 30°=1/2, sin 45°=√2/2, sin 60°=√3/2
- Double angle: sin 2θ = 2 sin θ cos θ; cos 2θ = cos²θ - sin²θ
- Sum angle: sin(A+B) = sin A cos B + cos A sin B
The domain of tan θ is undefined at θ = 90° + 180°k (where cos θ = 0).
💡 Worked example: Calculate sin, cos, tan for θ = 60°
Given:- θ = 60° = π/3 rad
- Special angle of the 30-60-90 triangle
Steps:- Convert to radians: 60° × π/180 = π/3 ≈ 1.0472 rad.
- sin 60° = √3/2 ≈ 0.8660.
- cos 60° = 1/2 = 0.5.
- tan 60° = sin 60° / cos 60° = (√3/2)/(1/2) = √3 ≈ 1.7321.
- Verify Pythagorean: sin² + cos² = 3/4 + 1/4 = 1 ✓.
Result: sin 60° = √3/2 ≈ 0.866; cos 60° = 0.5; tan 60° = √3 ≈ 1.732.
❓ Frequently asked questions
When should I use degrees vs. radians?
Degrees are more intuitive for everyday geometry, land surveying, navigation (GPS), and astronomy (RA/Dec). Radians are required for calculus because the derivative of sin x = cos x only holds in radians; they are also used for physics (harmonic motion, waves), engineering (Fourier analysis), and programming (Math.sin in JavaScript uses radians). Conversion: π rad = 180°.
How do I memorize special angle values?
The 'left hand' trick: count 1, 2, 3, 4 for angles 0°, 30°, 45°, 60°. sin θ = √(finger count)/2; for example, √1/2 = 1/2 for 30°. Or memorize the triangles: 30-60-90 (sides 1:√3:2) and 45-45-90 (sides 1:1:√2). From these, derive everything: sin 30° = 1/2, sin 45° = √2/2, sin 60° = √3/2.
What is the difference between trig and inverse trig functions?
Direct functions: sin θ → ratio (0-1 for acute angles). Inverse functions (arcsin, sin⁻¹, asin): ratio → angle. Because sin is not one-to-one, the range of arcsin is restricted to [-90°, 90°] and arccos to [0°, 180°] to keep the function well-defined. Applications: surveyors use arctan to calculate slope grade from rise/run ratios; arccos determines the angle between two vectors.
What are real-life applications of trigonometry?
Surveying & GIS - measuring mountain heights from elevation angle and distance (h = d × tan θ); architecture - roof and stair design; astronomy - stellar parallax; acoustics - sound waves; physics - projectile motion (R = v² sin 2θ / g); electronics - AC current analysis; computer graphics - sprite rotation; navigation - the haversine formula for distance between two GPS points.
Why is tan θ undefined at 90°?
Because tan θ = sin θ / cos θ, and cos 90° = 0, which causes division by zero. Graphically, tan has vertical asymptotes at θ = 90° + 180°k (e.g., 90°, 270°, etc.). As θ approaches 90° from the left, tan → +∞; from the right, tan → -∞. The calculator will show an error or 'undefined'. The same applies to sec θ and cot θ at angles where the denominator is zero.
📚 Sources & references
Last updated: May 11, 2026