11.1 Angles and Radian Measure

Degree vs Radian Measures

Degrees and radians are two ways to measure angles. Degrees are familiar from geometry, but radians connect angle measurement directly to the geometry of a circle. Most of the formulas you'll use in trigonometry require angles in radians, so getting comfortable with both systems and converting between them is essential.

Converting Between Degrees and Radians

A full circle contains 360° or $2\pi$ radians. That relationship gives you two conversion factors:

• Radians → Degrees: Multiply by $\frac{180}{\pi}$
  • Example: $\frac{\pi}{4} \times \frac{180}{\pi} = 45°$
• Degrees → Radians: Multiply by $\frac{\pi}{180}$
  • Example: $60° \times \frac{\pi}{180} = \frac{\pi}{3}$ radians

Memorize these common conversions, since they show up constantly:

What a Radian Actually Means

A radian is defined as the angle created when the arc length equals the radius of the circle. If you have a circle with radius 5 and you measure an arc that's also 5 units long, the central angle is exactly 1 radian (roughly 57.3°).

This is why radians are so useful: they tie the angle directly to the arc length, which makes formulas much cleaner.

• Counterclockwise angles are positive
• Clockwise angles are negative

Angles and the Unit Circle

Unit Circle Basics

The unit circle is a circle with radius 1, centered at the origin $(0, 0)$. Because the radius is 1, the circumference is $2\pi$, which means one full counterclockwise rotation equals $2\pi$ radians.

Angles on the unit circle are measured from the positive x-axis, increasing counterclockwise. Any point on the unit circle can be written as $(\cos\theta, \sin\theta)$, which is how the trig functions get their geometric meaning.

Key Points on the Unit Circle

These four points sit at the axes and are worth memorizing:

• $(1, 0)$ at $0$ radians (0°)
• $(0, 1)$ at $\frac{\pi}{2}$ radians (90°)
• $(-1, 0)$ at $\pi$ radians (180°)
• $(0, -1)$ at $\frac{3\pi}{2}$ radians (270°)

For angles between the axes, the coordinates give you sine and cosine directly. For example, at $\frac{\pi}{4}$ (45°), the point on the unit circle is $\left(\frac{\sqrt{2}}{2},\, \frac{\sqrt{2}}{2}\right)$, so $\cos\frac{\pi}{4} = \frac{\sqrt{2}}{2}$ and $\sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}$.

Arc Length and Sector Area

These two formulas are direct applications of radian measure. Both require the angle to be in radians, so always convert first if you're given degrees.

Arc Length

Arc length is the distance along the curve of a circle between two points. The formula is:

$s = r\theta$

where $s$ is the arc length, $r$ is the radius, and $\theta$ is the central angle in radians.

Example: Find the arc length for a circle with radius 6 and a central angle of $\frac{\pi}{3}$.

1. Confirm the angle is in radians. $\frac{\pi}{3}$ is already in radians.
2. Substitute into the formula: $s = 6 \times \frac{\pi}{3}$
3. Simplify: $s = 2\pi \approx 6.28$ units

Sector Area

A sector is the "pie slice" region between two radii and their connecting arc. The formula is:

$A = \frac{1}{2}r^2\theta$

where $A$ is the sector area, $r$ is the radius, and $\theta$ is the central angle in radians.

Example: Find the sector area for a circle with radius 4 and a central angle of $\frac{\pi}{6}$.

1. Confirm the angle is in radians. $\frac{\pi}{6}$ is already in radians.
2. Substitute: $A = \frac{1}{2}(4)^2 \times \frac{\pi}{6}$
3. Simplify: $A = \frac{1}{2}(16) \times \frac{\pi}{6} = \frac{8\pi}{3} \approx 8.38$ square units

Common Mistakes to Watch For

• Using degrees in the formula. If you plug 60° directly into $s = r\theta$, you'll get a wildly wrong answer. Convert to radians first.
• Mixing up the formulas. Arc length is $r\theta$ (linear, so no squaring). Sector area is $\frac{1}{2}r^2\theta$ (area, so the radius is squared). Notice that sector area has the same structure as triangle area ($\frac{1}{2} \times \text{base} \times \text{height}$).
• Forgetting units. Arc length is in linear units (cm, ft). Sector area is in square units (cm², ft²).