Trigonometry

Trigonometry connects angles to lengths. On the unit circle (radius 1), a point at angle θ has coordinates exactly (cos θ, sin θ). That single fact powers 2D/3D rotation, robot kinematics, and the waves behind sound and signals.

Drag the angle and watch the point travel around the circle. The gold segment is cos θ (horizontal), the green segment is sin θ (vertical).

angle θ = 45°

cos θ = 0.707sin θ = 0.707tan θ = 1.000

The three ratios

For an angle in a right triangle:

• sin θ = opposite / hypotenuse
• cos θ = adjacent / hypotenuse
• tan θ = sin θ / cos θ = opposite / adjacent

On the unit circle the hypotenuse is 1, so the point’s height is sin θ and its horizontal position is cos θ. As θ sweeps 0 → 360°, both trace smooth waves.

Radians

Angles are often measured in radians instead of degrees: a full circle is 2π radians (≈ 6.28), so 180° = π. Radians make calculus and code cleaner — most programming sin/cos functions expect radians.

Where it shows up in CS

• Graphics & games: rotating and positioning objects.
• Robotics: a joint at angle θ puts the arm tip at (cos θ, sin θ) distances.
• Signals & audio: every sound decomposes into sine waves (Fourier analysis).

Going deeper

Khan Academy: Trigonometry is a free, thorough course.

Takeaways

• The unit circle ties angle θ directly to (cos θ, sin θ).
• sin/cos/tan are ratios; on the unit circle they are coordinates and slope.
• Rotations, robot arms, and signals all run on trigonometry.