IB Mathematics AA The circle radian measure of angles Study Notes
IB Mathematics AA The circle radian measure of angles Study Notes
IB Mathematics AA The circle radian measure of angles Study Notes Offer a clear explanation of The circle radian measure of angles , including various formula, rules, exam style questions as example to explain the topics. Worked Out examples and common problem types provided here will be sufficient to cover for topic The circle radian measure of angles.
The Circle — Radian Measure of Angles
The Circle – Radian Measure of Angles
A radian is the standard unit for measuring angles in mathematics, especially in trigonometry and calculus. One radian is the angle formed at the center of a circle by an arc whose length is equal to the radius of the circle.
Relation between Radians and Degrees
- The circumference of a circle is \(2 \pi r\).
- The full angle around a point is \(360^\circ\) or \(2 \pi\) radians.
- The conversion formula:
- \( 180^\circ = \pi \ \text{radians} \)
- \( 1 \ \text{radian} = \frac{180^\circ}{\pi} \approx 57.296^\circ \)
- \( 1^\circ = \frac{\pi}{180} \ \text{radians} \)
Example
A sector of a circle has a radius of 5 cm and the angle at the center is 72°. Find: The angle in radians
▶️Answer/Explanation
Convert angle to radians:
\( \theta = 72^\circ \times \frac{\pi}{180} = \frac{2\pi}{5} \approx 1.257 \ \text{radians} \)
Length of an Arc
Length of an Arc
The length of an arc of a circle is directly proportional to the angle it subtends at the center of the circle.
\( s = r \theta \)
Where:
\( s \) = arc length
\( r \) = radius of the circle
\( \theta \) = angle in radians
If the angle is given in degrees, first convert it to radians:
\( \theta = \text{degrees} \times \frac{\pi}{180} \)
Why use radians? Because the formula \( s = r \theta \) only works directly when \( \theta \) is in radians.
Example
Find the length of an arc of a circle with radius 10 cm that subtends an angle of 2 radians at the center.
▶️Answer/Explanation
Use the arc length formula: \( s = r \theta \)
Substitute the given values: \( s = 10 \times 2 = 20 \ \text{cm} \)
Area of a Sector
Area of a Sector
The area of a sector of a circle is proportional to the angle it subtends at the center.
\( A = \frac{1}{2} r^2 \theta \)
Where:
\( A \) = area of the sector
\( r \) = radius of the circle
\( \theta \) = angle in radians
If the angle is given in degrees, convert to radians first:
\( \theta = \text{degrees} \times \frac{\pi}{180} \)
Why radians? The formula works directly with radians to give the correct area.
Example
A sector of a circle has a radius of 7 cm and subtends an angle of 60° at the center. Find its area.
▶️Answer/Explanation
Convert angle to radians:
\( \theta = 60^\circ \times \frac{\pi}{180} = \frac{\pi}{3} \approx 1.047 \ \text{radians} \)
Apply the area formula:
\( A = \frac{1}{2} \times 7^2 \times 1.047 = \frac{1}{2} \times 49 \times 1.047 = 25.6 \ \text{cm}^2 \)