Angles and Their Measure: Degrees, Radians, Arc Length, and Sector Area

Angles and Their Measure

Introduction to Angles

Angles are fundamental geometric objects formed by two rays sharing a common endpoint called the vertex. In trigonometry and precalculus, angles are used to describe rotation and position in a plane.

• Vertex: The common endpoint of the two rays.

• Initial Side: The starting position of the ray.

• Terminal Side: The position of the ray after rotation.

• Positive Angle: Formed by counterclockwise rotation from the initial side.

• Negative Angle: Formed by clockwise rotation from the initial side.

Example: Drawing a 45° angle involves rotating the terminal side 45° counterclockwise from the initial side.

Angles in Standard Position

An angle is in standard position if its vertex is at the origin of a coordinate plane and its initial side lies along the positive x-axis.

• Quadrantal Angles: Angles whose terminal side lies on the x-axis or y-axis (e.g., 0°, 90°, 180°, 270°).

• Angles in Quadrants: The terminal side of an angle may lie in any of the four quadrants, depending on its measure.

Example: A 135° angle has its terminal side in the second quadrant.

Measuring Angles: Degrees and Radians

Degrees

The degree is a unit of angular measure. One full revolution is 360 degrees.

• 1 revolution = 360°

• Quadrantal Angles: 0°, 90°, 180°, 270°, 360°

Example: A 90° angle is a right angle, with its terminal side along the positive y-axis.

Radians

The radian is another unit of angular measure, based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc equal in length to the radius.

• 1 revolution = 2π radians

• Relationship: $180^ ext{o} = ext{π radians}$

Example: An angle of π/2 radians is equivalent to 90°.

Converting Between Degrees and Radians

To convert between degrees and radians, use the following formulas:

• Degrees to Radians: $ext{radians} = ext{degrees} imes rac{ ext{π}}{180}$

• Radians to Degrees: $ext{degrees} = ext{radians} imes rac{180}{ ext{π}}$

Example: Convert 150° to radians:

$150^ ext{o} imes rac{ ext{π}}{180} = rac{5 ext{π}}{6}$ radians

Example: Convert $\frac{7\pi}{6}$ radians to degrees:

$\frac{7\pi}{6} \times \frac{180}{\pi} = 210^ ext{o}$

Common Degree-Radian Conversions Table

Degrees	Radians
30°	$\frac{\pi}{6}$
45°	$\frac{\pi}{4}$
60°	$\frac{\pi}{3}$
90°	$\frac{\pi}{2}$
120°	$\frac{2\pi}{3}$
135°	$\frac{3\pi}{4}$
150°	$\frac{5\pi}{6}$
180°	$\pi$

Arc Length and Area of a Sector

Arc Length

The length of an arc of a circle is proportional to the radius and the angle (in radians) subtended by the arc.

• Formula: $s = r\theta$

• s: Arc length

• r: Radius of the circle

• θ: Central angle in radians

Example: Find the arc length of a circle with radius 4.05 m and central angle 0.87 radians:

$s = 4.05 \times 0.87 = 3.52$ meters

Area of a Sector

The area of a sector of a circle is a fraction of the area of the entire circle, determined by the central angle (in radians).

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

• A: Area of the sector

• r: Radius of the circle

• θ: Central angle in radians

Example: Find the area of a sector with radius 6 inches and central angle 30° (convert to radians first):

$\theta = 30^ ext{o} \times \frac{\pi}{180} = \frac{\pi}{6}$

$A = \frac{1}{2} \times 6^2 \times \frac{\pi}{6} = 3\pi$ square inches

Practice Problems

• Draw angles of various measures in standard position.

• Convert between degrees and radians.

• Find arc length and area of a sector for given radius and angle.

Summary Table: Key Formulas

Concept	Formula
Degrees to Radians	$\text{radians} = \text{degrees} \times \frac{\pi}{180}$
Radians to Degrees	$\text{degrees} = \text{radians} \times \frac{180}{\pi}$
Arc Length	$s = r\theta$
Area of Sector	$A = \frac{1}{2} r^2 \theta$

Additional info: These notes cover foundational trigonometric concepts from Precalculus Chapter 5 (Trigonometric Functions), specifically the measurement of angles, conversion between units, and geometric applications involving circles.