Angles, Unit Circle, and Radian Measure

Angles and Their Measurement

Standard Position and Terminology

An angle is formed by rotating a ray (the initial side) about its endpoint (the vertex) to a new position (the terminal side). Angles are fundamental in trigonometry and precalculus, especially when analyzing rotation and periodic phenomena.

• Standard Position: An angle is in standard position if its vertex is at the origin and its initial side lies along the positive x-axis.

• Positive Angles: Generated by counterclockwise rotation.

• Negative Angles: Generated by clockwise rotation.

• Greek Letters: Common symbols for angles include θ, φ, α, β.

Quadrants and Angle Classification

The coordinate plane is divided into four quadrants. The position of an angle's terminal side determines its quadrant:

• Quadrant I: 0° < θ < 90°

• Quadrant II: 90° < θ < 180°

• Quadrant III: 180° < θ < 270°

• Quadrant IV: 270° < θ < 360°

Complementary and Supplementary Angles

• Complementary Angles: Two angles whose measures add up to 90°.

• Supplementary Angles: Two angles whose measures add up to 180°.

• Example: 30° and 60° are complementary; 120° and 60° are supplementary.

Coterminal Angles

Coterminal angles share the same initial and terminal sides but may have different measures. They can be found by adding or subtracting full rotations (360° or $2\pi$ radians):

• Formula (degrees): $\theta + 360^\circ \cdot n$

• Formula (radians): $\theta + 2\pi n$

• Example: 45° and 405° are coterminal because $405^\circ = 45^\circ + 360^\circ$.

Degree and Radian Measure

Degrees

• A degree is 1/360 of a full rotation.

• There are 360° in a circle.

Radians

• A radian is the angle subtended at the center of a circle by an arc whose length equals the radius of the circle.

• There are $2\pi$ radians in a full circle.

• Conversion: $360^\circ = 2\pi$ radians

• Formulas:

  • Degrees to radians: $\theta_{\text{rad}} = \theta_{\text{deg}} \cdot \frac{\pi}{180}$

  • Radians to degrees: $\theta_{\text{deg}} = \theta_{\text{rad}} \cdot \frac{180}{\pi}$

• Example: Convert 120° to radians:

  • $120^\circ \cdot \frac{\pi}{180} = \frac{2\pi}{3}$ radians

• Example: Convert $\frac{5\pi}{4}$ radians to degrees:

  • $\frac{5\pi}{4} \cdot \frac{180}{\pi} = 225^\circ$

The Unit Circle

Definition and Equation

The unit circle is a circle centered at the origin with radius 1. Its equation is:

• $x^2 + y^2 = 1$

Coordinates on the Unit Circle

• Any point $(x, y)$ on the unit circle satisfies $x^2 + y^2 = 1$.

• Common points:

  • $(\frac{\sqrt{3}}{2}, \frac{1}{2})$ at 30° ($\frac{\pi}{6}$ radians)

  • $(\frac{1}{2}, \frac{\sqrt{3}}{2})$ at 60° ($\frac{\pi}{3}$ radians)

  • $(\frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}})$ at 45° ($\frac{\pi}{4}$ radians)

• To check if a point is on the unit circle, substitute into $x^2 + y^2 = 1$.

• Example: Is $(\frac{3}{4}, \frac{1}{4})$ on the unit circle?

  • $(\frac{3}{4})^2 + (\frac{1}{4})^2 = \frac{9}{16} + \frac{1}{16} = \frac{10}{16} = \frac{5}{8} \neq 1$

  • So, it is not on the unit circle.

Arc Length and Sector Area

Arc Length

• The arc length $s$ of a circle of radius $r$ subtended by a central angle $\theta$ (in radians) is:

$s = r\theta$

• Example: For $r = 7$, $\theta = \frac{\pi}{2}$, $s = 7 \cdot \frac{\pi}{2} = \frac{7\pi}{2}$

Sector Area

• The area $A$ of a sector of a circle of radius $r$ and central angle $\theta$ (in radians) is:

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

• Example: For $r = 7$, $\theta = \frac{\pi}{2}$, $A = \frac{1}{2} \cdot 7^2 \cdot \frac{\pi}{2} = \frac{49\pi}{4}$

Summary Table: Common Angles and Their Radian Measures

Degrees (°)	Radians	Quadrant
30	$\frac{\pi}{6}$	I
45	$\frac{\pi}{4}$	I
60	$\frac{\pi}{3}$	I
90	$\frac{\pi}{2}$	I/II
120	$\frac{2\pi}{3}$	II
135	$\frac{3\pi}{4}$	II
180	$\pi$	II/III
225	$\frac{5\pi}{4}$	III
240	$\frac{4\pi}{3}$	III
270	$\frac{3\pi}{2}$	III/IV
360	$2\pi$	I

Additional info: Some context and examples were inferred to clarify the use of the unit circle, angle classification, and formulas for arc length and sector area, as well as to provide a summary table of common angles.