Key Concepts:
Radians and Degrees:
- A radian is an alternative unit for measuring angles, where one full revolution (360°) equals \(2\pi\) radians. - The conversion formulas between degrees and radians are: \[ 1^\circ = \frac{\pi}{180} \text{ radians}, \quad 1 \text{ radian} = \frac{180}{\pi}^\circ. \]Arc Length:
- The length of an arc of a circle is given by: \[ l = r\theta, \] where \(r\) is the radius and \(\theta\) is the central angle in radians.Sector Area:
- The area of a sector of a circle is given by: \[ A = \frac{1}{2} r^2 \theta, \] where \(r\) is the radius and \(\theta\) is the central angle in radians.Applications:
- Circular measure is used in physics, engineering, and navigation for measuring distances along circular paths.Examples:
Example 1: Convert \(135^\circ\) to radians.
Solution:
Using the conversion formula: \[ \theta = 135^\circ \times \frac{\pi}{180}. \] \[ \theta = \frac{135\pi}{180}. \] \[ \theta = \frac{3\pi}{4} \text{ radians}. \] Thus, \(135^\circ = \frac{3\pi}{4}\) radians.Example 2: Convert \(2.5\) radians to degrees.
Solution:
Using the conversion formula: \[ \theta = 2.5 \times \frac{180}{\pi}. \] \[ \theta = \frac{450}{\pi} \approx 143.24^\circ. \] Thus, \(2.5\) radians \(\approx 143.2^\circ\).Example 3: Find the length of an arc in a circle of radius \(10\) cm subtended by a central angle of \(1.2\) radians.
Solution:
Using the arc length formula: \[ l = r\theta. \] Substituting values: \[ l = 10 \times 1.2 = 12 \text{ cm}. \] Thus, the arc length is \(12\) cm.Example 4: Find the area of a sector with radius \(8\) cm and central angle \(2\) radians.
Solution:
Using the sector area formula: \[ A = \frac{1}{2} r^2 \theta. \] Substituting values: \[ A = \frac{1}{2} \times 8^2 \times 2. \] \[ A = \frac{1}{2} \times 64 \times 2 = 64 \text{ cm}^2. \] Thus, the area of the sector is \(64\) cm².