Unit 4: Circle Trigonometry – Angles, Radians, and Arc Length

Circle Trigonometry

Angle Measurement

When we measure an angle, we are quantifying the amount of rotation from an initial side to a terminal side. This measurement is independent of the size of the circle and can be expressed in degrees or radians.

• Angle Measure: Represents the fraction of a full rotation about a point (the center of a circle).

• Arc Length (s): The distance along the circumference of a circle that is subtended by a central angle. It is a portion of the total circumference.

• Sector Area (As): The area enclosed by two radii and the arc between them. It is a portion of the total area of the circle.

Key Formulas:

• Circumference of a circle: $C = 2\pi r$

• Area of a circle: $A = \pi r^2$

Degree and Radian Measure

Angles can be measured in degrees or radians. Radians are the standard unit in higher mathematics and are based on the radius of the circle.

• Degree: A full rotation is $360^\circ$.

• Radian: One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius of the circle.

• Conversion:

  • $180^\circ = \pi$ radians

  • To convert degrees to radians: $\text{radians} = \text{degrees} \times \frac{\pi}{180}$

  • To convert radians to degrees: $\text{degrees} = \text{radians} \times \frac{180}{\pi}$

Example: Convert $90^\circ$ to radians:

• $90^\circ \times \frac{\pi}{180} = \frac{\pi}{2}$ radians

Example: Convert $2.7$ radians to degrees:

• $2.7 \times \frac{180}{\pi} \approx 154.7^\circ$

Arc Length

The arc length is the distance along the circle's edge between two points, defined by a central angle $\theta$ (in radians).

• Formula: $s = r\theta$, where $s$ is arc length, $r$ is radius, and $\theta$ is in radians.

Example: If $r = 2$ inches and $\theta = 0.45$ radians, then $s = 2 \times 0.45 = 0.9$ inches.

Sector Area

The area of a sector (a 'slice' of the circle) is proportional to the angle it subtends at the center.

• Formula: $A_s = \frac{1}{2} r^2 \theta$, where $\theta$ is in radians.

Example: For $r = 3$ and $\theta = 1$ radian, $A_s = \frac{1}{2} \times 3^2 \times 1 = 4.5$ square units.

Understanding Radians Visually

• One radian is the angle that cuts off an arc equal in length to the radius.

• A full circle is $2\pi$ radians ($\approx 6.28$ radians).

• Half a circle is $\pi$ radians ($180^\circ$).

Table: Fractional Parts of a Circle

Portion of Circle	Degrees	Radians
1 (full circle)	360°	$2\pi$
1/2	180°	$\pi$
1/4	90°	$\frac{\pi}{2}$

Summary

• Angles can be measured in degrees or radians, with radians being the standard in advanced mathematics.

• Arc length and sector area are directly related to the angle in radians.

• Conversion between degrees and radians is essential for solving trigonometric problems.

Additional info: The notes emphasize the geometric meaning of radians and their relationship to arc length and sector area, which are foundational for understanding trigonometric functions and their applications in Precalculus.