Trigonometric Ratios
Understanding sine, cosine, tangent, and other trigonometric ratios
Interactive Trigonometric Ratios Visualization
Welcome to the Trigonometric Ratios Explorer
This interactive visualization helps you understand trigonometric ratios (sine, cosine, tangent, and their reciprocals) through both the unit circle and right triangle perspectives. Explore how these ratios change as the angle varies.
What you can explore:
- Unit Circle View - See how sin(θ) and cos(θ) relate to coordinates on the unit circle
- Right Triangle View - Understand ratios as opposite/hypotenuse, adjacent/hypotenuse, and opposite/adjacent
- All Six Ratios - Sine, cosine, tangent, and their reciprocals (cosecant, secant, cotangent)
- Quadrant Information - See which quadrant the angle is in and sign of each ratio
- Reference Angles - Understand how angles relate to their acute reference angles
How to Use This Visualization
Interactive Features:
- • Adjust Angle - Use the slider to change the angle from 0° to 360°
- • Switch Views - Toggle between Unit Circle and Right Triangle visualizations
- • Toggle Labels - Show/hide ratio labels and side names
- • View Ratios - See all six trigonometric ratios with their values
- • Quadrant Info - Display quadrant information and sign analysis
What You'll See:
- • Unit Circle - Circle with radius 1 showing sin(θ) and cos(θ) as coordinates
- • Right Triangle - Triangle showing opposite, adjacent, and hypotenuse sides
- • Cosine Line (red) - Horizontal distance from center (x-coordinate)
- • Sine Line (amber) - Vertical distance from center (y-coordinate)
- • Angle Arc (purple) - Visual representation of the angle
- • Ratio Values - Real-time calculation of all trigonometric ratios
Adjust to see how ratios change
Trigonometric Ratios
Sine
sin(30°) = 0.5000
y-coordinate
Cosine
cos(30°) = 0.8660
x-coordinate
Tangent
tan(30°) = 0.5774
sin/cos
Cosecant
csc(30°) = 2.0000
1/sin(θ)
Secant
sec(30°) = 1.1547
1/cos(θ)
Cotangent
cot(30°) = 1.7321
1/tan(θ)
Angle Information
Angle: 30° (0.5236 rad)
Reference: 30.00°
Quadrant: 1
Key Concepts
Unit Circle: On a unit circle, cos(θ) is the x-coordinate and sin(θ) is the y-coordinate of the point where the terminal side intersects the circle.
Right Triangle: In a right triangle, sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent.
Signs: In Q1: all positive, Q2: sin positive, Q3: tan positive, Q4: cos positive (SOH-CAH-TOA mnemonic: All-Sin-Tan-Cos).
Trigonometric Ratios Definitions
Primary Ratios:
- sin(θ) = opposite/hypotenuse
- cos(θ) = adjacent/hypotenuse
- tan(θ) = opposite/adjacent
Reciprocal Ratios:
- csc(θ) = 1/sin(θ)
- sec(θ) = 1/cos(θ)
- cot(θ) = 1/tan(θ)
Lessons
Individual learning units
Lessons coming soon