Angles, Radian Measure, and Arc Length

Introduction to Angles

An angle is formed by rotating a ray around its endpoint. The starting position is the initial side, and the ending position is the terminal side. The endpoint is the vertex.

• Positive Angle: The rotation is counter-clockwise.

• Negative Angle: The rotation is clockwise.

Systems of Angle Measurement

1. Degree Measure (Sexagesimal System):

   This is the most common system in general use. One full counter-clockwise revolution is divided into 360 parts, each called a degree.

   • 1 full revolution = $360^\circ$

   • 1 degree ($1^\circ$) = 60 minutes ($60'$)

   • 1 minute ($1'$) = 60 seconds ($60''$)

2. Radian Measure (Circular System):

   This is the standard system for calculus and higher mathematics because it relates the angle directly to a length.

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

Relationship Between Degrees and Radians

The circumference of a circle is $2\pi r$. The number of radians in one full circle is the circumference divided by the radius, which is $\frac{2\pi r}{r} = 2\pi$. Since one full revolution is also $360^\circ$, we have the fundamental relationship:

$$2\pi \text{ radians} = 360^\circ \implies \pi \text{ radians} = 180^\circ$$

• Conversion Formulas:

  • To convert from degrees to radians, multiply by $\frac{\pi}{180}$.

  • To convert from radians to degrees, multiply by $\frac{180}{\pi}$.

Arc Length and Area of a Sector

For a circle of radius 'r', and a central angle '$\theta$' that must be in radians:

• Arc Length (l): The length of the arc intercepted by the angle $\theta$.

$$l = r\theta$$

• Area of Sector (A): The area of the 'slice of pie' created by the angle $\theta$.

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

Example 1: Degree to Radian Conversion

Question: Convert $40^\circ 20'$ into radian measure.

Solution:

Step 1: Convert minutes to decimal degrees. $20' = (\frac{20}{60})^\circ = (\frac{1}{3})^\circ$.

Step 2: Add this to the degrees part: $40^\circ + (\frac{1}{3})^\circ = 40 \frac{1}{3}^\circ = \frac{121}{3}^\circ$.

Step 3: Convert the total degrees to radians: Radians = Degrees $\times \frac{\pi}{180} = \frac{121}{3} \times \frac{\pi}{180} = \mathbf{\frac{121\pi}{540}}$ radians.

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Example 2: Radian to Degree Conversion

Question: Convert 6 radians into degree measure.

Solution:

Step 1: Use the conversion formula. Degrees = Radians $\times \frac{180}{\pi} = 6 \times \frac{180}{22/7} = \frac{6 \times 180 \times 7}{22} = \frac{3780}{11}$ degrees.

Step 2: Convert the improper fraction to a mixed fraction. $3780 \div 11 = 343$ with a remainder of 7. So, we have $343 \frac{7}{11}^\circ$.

Step 3: Convert the fractional part to minutes. $\frac{7}{11}^\circ = \frac{7}{11} \times 60' = \frac{420}{11}' = 38 \frac{2}{11}'$. We now have $343^\circ 38' \frac{2}{11}''$.

Step 4: Convert the remaining fractional part to seconds. $\frac{2}{11}' = \frac{2}{11} \times 60'' \approx 10.9'' \approx 11''$.

The angle is approximately $343^\circ 38' 11''$.

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Example 3: Arc Length Calculation

Question: Find the length of an arc of a circle of radius 5 cm subtending a central angle of $15^\circ$.

Solution:

Step 1: The formula $l = r\theta$ requires the angle to be in radians. Convert $15^\circ$ to radians:

$\theta = 15 \times \frac{\pi}{180} = \frac{\pi}{12}$ radians.

Step 2: Use the arc length formula:

$l = r\theta = 5 \times \frac{\pi}{12} = \mathbf{\frac{5\pi}{12}}$ cm.

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Example 4: Clock Angle Problem

Question: Find the angle between the minute hand and the hour hand of a clock when the time is 7:20 AM.

Solution:

We measure angles clockwise from the 12 o'clock position.

• Minute Hand Speed: $360^\circ$ in 60 min = $6^\circ$ per minute.

• Hour Hand Speed: $360^\circ$ in 12 hours (720 min) = $0.5^\circ$ per minute.

Position of Minute Hand: At 20 minutes past the hour, its position is $20 \text{ min} \times 6^\circ/\text{min} = \mathbf{120^\circ}$.

Position of Hour Hand: At 7:20, it has been $7 \times 60 + 20 = 440$ minutes past 12. Its position is $440 \text{ min} \times 0.5^\circ/\text{min} = \mathbf{220^\circ}$.

Angle Between Hands: The difference in their positions is $|220^\circ - 120^\circ| = \mathbf{100^\circ}$.