7.1 Angles

Angles and circular motion are key concepts in math, linking geometry and trigonometry. They help you understand how objects move in circles and how to measure rotations, which comes up in physics, engineering, and throughout trigonometry.

This section covers different ways to measure angles, how to convert between units, how to calculate arc lengths, and the basics of circular motion (linear and angular velocity).

Angles and Circular Motion

Angles in standard position

An angle is in standard position when its vertex sits at the origin and its initial side extends along the positive x-axis. The other ray, called the terminal side, is where the angle "ends up" after rotation.

• Positive angles are measured counterclockwise from the initial side (e.g., 90°, 180°)
• Negative angles are measured clockwise from the initial side (e.g., -45°, -120°)

A common mistake is mixing up the direction. Think of it this way: counterclockwise = positive, clockwise = negative.

Quadrantal angles are special cases where the terminal side lands exactly on one of the coordinate axes:

• 0° or 0 radians: positive x-axis
• 90° or $\frac{\pi}{2}$ radians: positive y-axis
• 180° or $\pi$ radians: negative x-axis
• 270° or $\frac{3\pi}{2}$ radians: negative y-axis

Degree and radian conversions

Degrees and radians are two different units for measuring angles, just like inches and centimeters both measure length. One radian is the angle you get when the arc length along the circle equals the radius of that circle. A full circle is 360° or $2\pi$ radians.

Degrees → Radians: multiply by $\frac{\pi}{180}$

• 60° = $60 \times \frac{\pi}{180} = \frac{\pi}{3}$ radians
• 135° = $135 \times \frac{\pi}{180} = \frac{3\pi}{4}$ radians

Radians → Degrees: multiply by $\frac{180}{\pi}$

• $\frac{\pi}{6}$ radians = $\frac{\pi}{6} \times \frac{180}{\pi} = 30°$
• $\frac{5\pi}{4}$ radians = $\frac{5\pi}{4} \times \frac{180}{\pi} = 225°$

Notice that the $\pi$ cancels when you go from radians to degrees. That's a good way to check your work.

Coterminal angles

Coterminal angles share the same terminal side but differ by one or more full rotations. Visually, they "point" in the same direction.

In degrees: add or subtract multiples of 360°

• 45° and 405° are coterminal ($45° + 360° = 405°$)
• -30° and 330° are coterminal ($-30° + 360° = 330°$)

In radians: add or subtract multiples of $2\pi$

• $\frac{\pi}{3}$ and $\frac{7\pi}{3}$ are coterminal ($\frac{\pi}{3} + 2\pi = \frac{7\pi}{3}$)
• $-\frac{\pi}{4}$ and $\frac{7\pi}{4}$ are coterminal ($-\frac{\pi}{4} + 2\pi = \frac{7\pi}{4}$)

To find a positive coterminal angle for a negative angle, just keep adding 360° (or $2\pi$) until you get a positive result.

Arc length calculation

Arc length is the distance along the curved edge of a circle between two points. It's proportional to the central angle: a bigger angle sweeps out a longer arc.

Formula: $s = r\theta$

• $s$ = arc length
• $r$ = radius of the circle
• $\theta$ = central angle in radians (you must convert from degrees first if needed)

Example 1: Circle with radius 8 units and a central angle of 90°

1. Convert to radians: $90° \times \frac{\pi}{180} = \frac{\pi}{2}$
2. Apply the formula: $s = 8 \times \frac{\pi}{2} = 4\pi \approx 12.57$ units

Example 2: Circle with radius 3 units and a central angle of $\frac{2\pi}{3}$ radians

1. Already in radians, so apply directly: $s = 3 \times \frac{2\pi}{3} = 2\pi \approx 6.28$ units

The most common error here is forgetting to convert degrees to radians before plugging into the formula. If $\theta$ is in degrees, the formula won't give you the right answer.

Circular motion concepts

When an object moves along a circular path, you can describe its speed in two ways.

Linear velocity ($v$) is how fast the object travels along the circular path, measured in distance per unit time (like m/s).

$v = \frac{2\pi r}{T}$

• $r$ = radius of the circle
• $T$ = period (time for one complete revolution)

Angular velocity ($\omega$) is how fast the angle changes, measured in radians per unit time.

$\omega = \frac{2\pi}{T}$

These two quantities are connected by a clean relationship:

$v = r\omega$

This tells you that for the same angular velocity, a point farther from the center moves faster. Think of a merry-go-round: someone on the outer edge covers more distance per rotation than someone near the center.

Example 1: A Ferris wheel with radius 20 m completes one revolution in 30 seconds.

1. Linear velocity: $v = \frac{2\pi \times 20}{30} \approx 4.19$ m/s
2. Angular velocity: $\omega = \frac{2\pi}{30} \approx 0.21$ rad/s

Example 2: A car wheel with radius 0.3 m spins at 10 rad/s.

1. Linear velocity: $v = 0.3 \times 10 = 3$ m/s

Trigonometry and the Unit Circle

This section introduces a few foundational terms you'll use throughout the rest of the unit.

• Trigonometry is the branch of math dealing with relationships between the sides and angles of triangles.
• Unit circle: a circle with radius 1 centered at the origin. It's used to define sine, cosine, and the other trig functions for any angle, not just those in a triangle.
• Rotation is angular movement around a fixed point, measured in degrees or radians.
• Revolution is one complete rotation, equal to 360° or $2\pi$ radians.
• Circumference is the total distance around a circle: $C = 2\pi r$.
• Pi ($\pi$) is the ratio of any circle's circumference to its diameter, approximately 3.14159. This constant shows up everywhere in angle measurement because it connects straight-line distances to circular ones.