📐 Trigonometry Lab

Explore the unit circle, trig ratios, wave functions, and triangle laws — drag the angle to see it all live.

Click or drag anywhere on the canvas to set angle

🎛️ Angle

45°
π/4 rad

📊 Values

cos θ 0.707
sin θ 0.707
tan θ 1.000
csc θ 1.414
sec θ 1.414
cot θ 1.000
Amplitude: ×1.0
sin(x)
Period: 2π · Range: [−1, 1] · Odd function · sin(0)=0
cos(x)
Period: 2π · Range: [−1, 1] · Even function · cos(0)=1
tan(x)
Period: π · Range: (−∞, ∞) · Undefined at π/2 + nπ

⚖️ Law of Sines

a/sin(A) = b/sin(B) = c/sin(C)

Side a
Angle A (°)
Angle B (°)

📐 Law of Cosines

c² = a² + b² − 2ab·cos(C)

Side a
Side b
Angle C (°)

📏 Right Triangle (SOH CAH TOA)

Enter angle + one side

Angle θ (°)
Hypotenuse

🖼️ Triangle Diagram

Certain angles appear everywhere in math and physics. Memorise their exact values — they show up in calculus, physics, engineering, and exams.

AngleRadianssin θcos θtan θ
🔺 30-60-90 Triangle

Sides in ratio 1 : √3 : 2. If hypotenuse = 2, short leg = 1, long leg = √3.

🔷 45-45-90 Triangle

Sides in ratio 1 : 1 : √2. If legs = 1, hypotenuse = √2 ≈ 1.414.

🔄 CAST Rule

Quadrant I: All positive. II: Sin+. III: Tan+. IV: Cos+. Mnemonic: "All Students Take Calculus".

📐 Pythagorean Identity

sin²θ + cos²θ = 1 (always!). Also: 1+tan²θ = sec²θ and 1+cot²θ = csc²θ.

What is the unit circle?

A circle of radius 1 centred at the origin. For any angle θ, the point on the circle is (cos θ, sin θ). This makes sin and cos the x and y coordinates — a beautiful geometric definition.

SOH CAH TOA

For a right triangle: Sin = Opposite/Hypotenuse, Cos = Adjacent/Hypotenuse, Tan = Opposite/Adjacent. The unit circle generalises these to any angle.

Key identities
  • Pythagorean: sin²θ + cos²θ = 1
  • Angle sum: sin(A+B) = sinA cosB + cosA sinB
  • Double angle: sin(2θ) = 2sinθ cosθ
  • Law of Sines: a/sinA = b/sinB = c/sinC
  • Law of Cosines: c² = a² + b² − 2ab·cos(C)
Key terms
  • Radian — arc length on unit circle; 2π rad = 360°
  • Period — distance for a wave to repeat; sin and cos have period 2π
  • Amplitude — peak value; A·sin(x) has amplitude A
  • Phase shift — horizontal displacement; sin(x−π/2) = cos(x)
🎯 Try this challenge

Set the angle to 30°. What are sin, cos, and tan? Now try 150°. Why is sin(150°) = sin(30°)? Drag around the full circle and find all angles where sin = 0.5.

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