Angles, Arc Length and Sector Area
Angles, Arc Length and Sector Area
Basic Concepts
- Angles in trigonometry can be measured in degrees or radians; converting between the two is essential for solving problems.
- A radian is the measure of an angle that, when drawn as a central angle of a circle, intercepts an arc equal in length to the radius of the circle.
- 360 degrees is equivalent to 2π radians; 1 radian is approximately 57.3 degrees.
Radians and Degrees
- The formula for converting degrees to radians is: degrees = radians × (180/π).
- The formula for converting radians to degrees is: radians = degrees × (π/180).
Arc Length
- The arc length (s) of a sector of a circle with radius r and angle θ (in radians) is given by s = rθ.
Sector Area
- The sector area (A) of a circle with radius r and central angle θ (in radians) is given by A = 0.5r²θ.
Properties of Trigonometry and Circles
- Sin(θ) gives the y-coordinate of the point on the unit circle subtended by the angle θ.
- Cos(θ) gives the x-coordinate of the point on the unit circle subtended by the angle θ.
- The unit circle helps us to understand the cyclical nature of sine and cosine functions.
Solving Problems
- Switch comfortably between degrees and radians when solving problems.
- Learn to visualise problems related to angles, arcs, and sectors.
- Practice is key: solve a wide variety of problems dealing with angles, arc lengths, and sector areas to understand the interplay of these topics.