How do I compute trig functions for any angle?

Use the unit circle: (x, y) = (cos θ, sin θ). Find the reference angle in Q1, then apply the correct sign by quadrant (ASTC).

What are degrees vs radians?

Degrees: full circle = 360°. Radians: full circle = 2π. Convert with radians = degrees × π/180.

What is the reference angle?

The acute angle between the terminal side and the x-axis. Used to find exact values in any quadrant using Q1 values.

3 more

MATHEMATICSTrigonometryMathematics Calculator

🔢

Trigonometry

Compute all six trigonometric functions (sin, cos, tan, csc, sec, cot) for any angle. Quadrant, reference angle, Pythagorean identity check. Overview of trigonometry.

Start CalculatingExplore mathematical calculations

Examples — Click to Load

Angle (θ)

Unit

trig.sh

CALCULATED

$ trig --angle 30° --all

sin

0.5

cos

0.8660254

tan

0.57735027

Quadrant

csc

sec

1.15470054

cot

1.73205081

sin²+cos²

Trigonometry Result

sin = 0.5 | cos = 0.8660254

tan = 0.57735027

Q1sin²+cos² = 1

numbervibe.com/calculators/mathematics/trigonometry/trigonometry-calculator

All 6 Trig Functions

Function Values

sin² vs cos² (Pythagorean)

Calculation Breakdown

INPUT

Input Angle

30°

CONVERSION

Quadrant

30.0°

CONVERSION

Reference Angle

30°

PRIMARY RESULT

sin(θ)

0.5

y ext{on} ext{unit} circle

PRIMARY RESULT

cos(θ)

0.8660254

x ext{on} ext{unit} circle

tan(θ)

0.57735027

\text{sin}/\text{cos}

csc(θ)

1/\text{sin}

sec(θ)

1.15470054

1/\text{cos}

cot(θ)

1.73205081

\text{cos}/\text{sin}

VERIFICATION

sin²+cos²

ext{Pythagorean} ext{identity}

⚠️For educational and informational purposes only. Verify with a qualified professional.

Key Takeaways

• sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse in a right triangle
• tan = sin/cos, cot = cos/sin, sec = 1/cos, csc = 1/sin
• Quadrant I: all +; II: sin,csc +; III: tan,cot +; IV: cos,sec + (ASTC)
• sin²θ + cos²θ = 1 always (Pythagorean identity)
• On the unit circle: (cos θ, sin θ) is the point at angle θ

Did You Know?

📐Trigonometry means "triangle measurement" — it started with right triangles and was extended to all angles via the unit circleSource: Etymology

🌊All six trig functions appear in wave physics, AC circuits, and signal processing. Sine and cosine are the building blocksSource: Physics

🎮Game engines use sin and cos for rotations, circular motion, and camera control. Every 3D rotation uses trigSource: Game Development

📱Your phone uses trig for GPS, compass, accelerometer, and touch screen calculationsSource: Mobile Tech

🎵Music synthesis uses sine waves. Fourier showed any sound is a sum of sines and cosinesSource: Audio Engineering

🔭Astronomy uses spherical trigonometry for celestial coordinates and navigationSource: Astronomy

How It Works

Right Triangle Definition

In a right triangle with angle θ: sin θ = opposite/hypotenuse, cos θ = adjacent/hypotenuse, tan θ = opposite/adjacent. SOH-CAH-TOA.

Unit Circle Definition

For any angle θ, the point on the unit circle is (cos θ, sin θ). This extends trig to all angles. tan = y/x, sec = 1/x, csc = 1/y, cot = x/y. See Unit Circle Calculator.

Quadrant Signs

All positive in Q1. In Q2: sin,csc +. In Q3: tan,cot +. In Q4: cos,sec +. Use the reference angle and apply the sign from the quadrant.

Expert Tips

Memorize Special Angles

30°, 45°, 60°: sin = √1/2, √2/2, √3/2. cos is reversed. Use Unit Circle.

Reciprocal Functions

csc = 1/sin, sec = 1/cos, cot = 1/tan. When sin or cos = 0, the reciprocal is undefined.

Degrees vs Radians

360° = 2π rad. Calculus uses radians: d/dx[sin(x)] = cos(x) only when x is in radians.

Pythagorean Identity

sin²θ + cos²θ = 1 always. Use it to find one from the other. See Trig Identities.

Trig Calculator vs Other Tools

Feature	This Calculator	Scientific Calc	Manual
All 6 functions at once	✅	❌ One at a time	❌
Quadrant & reference	✅	❌	✅ Slow
Visual charts	✅	❌	❌
Step-by-step	✅	❌	✅
Pythagorean check	✅	❌	⚠️ Manual
Copy & share	✅	❌	❌
Degrees & radians	✅	✅	✅
Examples	✅	❌	❌

Frequently Asked Questions

What are the six trig functions?

Sine (sin), cosine (cos), tangent (tan), cosecant (csc = 1/sin), secant (sec = 1/cos), cotangent (cot = 1/tan). The first three are primary; the others are reciprocals.