Trigonometric Functions Calculator

Calculate sin, cos, tan, csc, sec, cot for any angle in degrees or radians. Get all six trig functions at once with step-by-step unit circle values and reference angle.

📐 Trigonometry📐 sin²(x) + cos²(x) = 1🔢 Math
Angle value
Unit
Unit
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Trigonometric Functions Calculator—sin²(x) + cos²(x) = 1dimensionless

Step-by-Step Examples

Example 1
30°

sin,cos,tan at 30°.

  • sin(30°)=0.5
  • cos(30°)=√3/2≈0.866
  • tan(30°)=1/√3≈0.577
✓ sin=0.5, cos=0.866, tan=0.577
Example 2
45°

sin,cos,tan at 45°.

  • sin=cos=√2/2≈0.707
  • tan=1
✓ sin=cos=0.707, tan=1
Example 3
90°

tan(90°) is undefined.

  • sin(90°)=1
  • cos(90°)=0
  • tan(90°)=undefined (division by zero)
✓ sin=1, cos=0, tan=undefined

Real-World Applications

🏗️
Engineering
Structural analysis, alternating current, wave analysis.
🎮
Computer Graphics
Rotation matrices use sin and cos for 2D/3D rotations.
🧬
Physics
Oscillations, waves, and electromagnetic fields modeled with trig functions.
📊
Signal Processing
Fourier series decomposes signals into sin and cos components.

Common Mistakes to Avoid

⚠️
Using degrees when calculator expects radians

Most programming languages (Python, JavaScript) use radians. Always convert: radians = degrees × π/180.

⚠️
tan undefined at 90°, 270°

cos(90°)=0 so tan=sin/cos is undefined. Not an error — it genuinely does not exist.

⚠️
sin²+cos²=1 always

This Pythagorean identity always holds. Use it to check your answers.

Frequently Asked Questions

What are the six trig functions? ▾
sin(opposite/hypotenuse), cos(adjacent/hypotenuse), tan(opposite/adjacent), and their reciprocals csc=1/sin, sec=1/cos, cot=1/tan.
What is SOHCAHTOA? ▾
Memory aid: Sin=Opposite/Hypotenuse, Cos=Adjacent/Hypotenuse, Tan=Opposite/Adjacent. For right triangle trig.
What are the key angles? ▾
0°,30°,45°,60°,90° have exact values. sin(30°)=½, cos(60°)=½, sin(45°)=cos(45°)=√2/2, tan(45°)=1, sin(60°)=√3/2.
What is the unit circle? ▾
Circle of radius 1. For angle θ, the point on unit circle is (cos θ, sin θ). Extends trig beyond right triangles to all angles.
What are trig function periods? ▾
sin and cos repeat every 360° (2π). tan repeats every 180° (π). sin and cos have range [−1,1]; tan has range (−∞,∞).

Related Math Calculators

📐 Inverse Trig📐 Degrees/Radians📐 Unit Circle📐 Law of Sines📐 Law of Cosines

Formula Explorer connections

Interpretation: This relationship connects dimensions, coordinates, angles or trigonometric ratios to a geometric measurement. Assumption: Use a consistent angle mode and length unit, identify the intended shape or coordinate system, and verify that the supplied dimensions form valid geometry.

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