Chapter 3: Trigonometry for Motion and Geometry

Use trigonometry as a practical tool for right triangles, motion, vectors, components, signs, and radians.

What You Will Practice

Trigonometry connects geometry to motion. It helps students describe angles, directions, components, and real-world systems used in physics, robotics, CAD, engineering technology, and calculus.

Right Triangles
Sine, Cosine, Tangent
Vector Components
Direction and Sign
Degrees to Radians
Radians to Degrees

Mini Lesson

1. Right Triangles

Right triangles are the foundation of basic trigonometry. The Pythagorean theorem connects the two legs and the hypotenuse.

a² + b² = c²

Example: If one leg is 6 and the hypotenuse is 10, then the other leg is 8.

2. Sine, Cosine, and Tangent

Sine, cosine, and tangent are side ratios in a right triangle.

sin(θ) = opposite / hypotenuse
cos(θ) = adjacent / hypotenuse
tan(θ) = opposite / adjacent

3. Resolving Components

Forces and motion at an angle can be split into horizontal and vertical components.

Fx = F cos(θ)
Fy = F sin(θ)

Example: A 20 N force at 30° has Fx ≈ 17.3 N and Fy = 10 N.

4. Direction and Sign

Direction matters. In a coordinate system, right and up are often positive; left and down are often negative.

Example: Motion downward usually has a negative vertical component.

5. Degrees and Radians

Degrees are common in geometry. Radians are required in calculus because they connect angles to arc length.

180° = π radians
90° = π/2 radians
45° = π/4 radians

Interactive Trigonometry Practice

Choose a topic and practice with instant feedback. Round component and decimal answers to two decimal places when needed.

Typing tip: For ratios, type either a fraction or decimal, like 3/5 or 0.6. For radians, type pi/2, pi/4, or pi.
Why this matters: Trigonometry is what lets students break forces, motion, robot movement, and angled geometry into understandable horizontal and vertical pieces.

Mastery Check

Before moving to Book 2 Chapter 4, students should be able to do the following.

Right Triangles

I can use the Pythagorean theorem to find a missing side.

Trig Ratios

I can calculate sine, cosine, and tangent from side lengths.

Components

I can resolve a force into horizontal and vertical components.

Signs

I can decide whether motion left/down is positive or negative.

Radians

I can convert common angles between degrees and radians.

Go to Chapter 4 Back to Book 2