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
Radian measure offers a natural and efficient way to measure angles in mathematics and physics. This topic explores the concept of radians, the length of an arc, and the area of a sector within a circle.
Key Concepts
Radian Measure of Angles:
- One radian is the angle subtended at the center of a circle by an arc equal in length to the circle’s radius.
- Conversion between radians and degrees:
- \( 1 \, \text{radian} = \frac{180}{\pi}^\circ \).
- \( 1^\circ = \frac{\pi}{180} \, \text{radians} \).
- Radians may be expressed as exact multiples of \( \pi \) (e.g., \( \frac{\pi}{3} \)) or as decimals.
Length of an Arc:
- \( \text{Arc Length} = r \theta \), where:
- \( r \): radius of the circle.
- \( \theta \): angle in radians.
- \( \text{Arc Length} = r \theta \), where:
Area of a Sector:
- \( \text{Sector Area} = \frac{1}{2} r^2 \theta \), where:
- \( r \): radius of the circle.
- \( \theta \): angle in radians.
- \( \text{Sector Area} = \frac{1}{2} r^2 \theta \), where:
Guidance, Clarifications, and Syllabus Links
- Radian measure is preferred in advanced mathematics and physics for its natural relationship to circular motion and periodic functions.
- Connections to physics:
- Diffraction patterns and circular motion rely on the use of radians for calculations.
- Historical context:
- Mathematicians like Seki Takakazu calculated \( \pi \) to ten decimal places.
- Explorations by Hipparchus, Menelaus, and Ptolemy contributed to the development of angle measurement.
- The division of a circle into 360 degrees originates from Babylonian mathematics.
- Theory of Knowledge (TOK):
- Which is a better measure of angle: radians or degrees?
- What criteria do mathematicians use to decide on the best system of measurement?
Examples
Example 1: Arc Length
- Find the length of an arc of a circle with radius \( 5 \, \text{cm} \) and subtending an angle of \( \frac{\pi}{3} \, \text{radians} \):
- \( \text{Arc Length} = r \theta = 5 \cdot \frac{\pi}{3} = \frac{5\pi}{3} \, \text{cm} \).
Example 2: Sector Area
- Find the area of a sector of a circle with radius \( 4 \, \text{cm} \) and angle \( \frac{\pi}{4} \, \text{radians} \):
- \( \text{Sector Area} = \frac{1}{2} r^2 \theta = \frac{1}{2} \cdot 4^2 \cdot \frac{\pi}{4} = \frac{8\pi}{4} = 2\pi \, \text{cm}^2 \).
IB Mathematics AA SL The circle radian measure of angles Exam Style Worked Out Questions
Question
A circular disc is cut into twelve sectors whose areas are in an arithmetic sequence.
The angle of the largest sector is twice the angle of the smallest sector.
Find the size of the angle of the smallest sector.
▶️Answer/Explanation
Markscheme
METHOD 1
If the areas are in arithmetic sequence, then so are the angles. (M1)
\( \Rightarrow {S_n} = \frac{n}{2}(a + l) \Rightarrow \frac{{12}}{2}(\theta + 2\theta ) = 18\theta \) M1A1
\( \Rightarrow 18\theta = 2\pi \) (A1)
\(\theta = \frac{\pi }{9}\) (accept \(20^\circ \)) A1
[5 marks]
METHOD 2
\({{\text{a}}_{12}} = 2{a_1}\) (M1)
\(\frac{{12}}{2}({a_1} + 2{a_1}) = \pi {r^2}\) M1A1
\(3{a_1} = \frac{{\pi {r^2}}}{6}\)
\(\frac{3}{2}{r^2}\theta = \frac{{\pi {r^2}}}{6}\) (A1)
\(\theta = \frac{{2\pi }}{{18}} = \frac{\pi }{9}\) (accept \(20^\circ \)) A1
[5 marks]
METHOD 3
Let smallest angle = a , common difference = d
\(a + 11d = 2a\) (M1)
\(a = 11d\) A1
\({S_n} = \frac{{12}}{2}(2a + 11d) = 2\pi \) M1
\(6(2a + a) = 2\pi \) (A1)
\(18a = 2\pi \)
\(a = \frac{\pi }{9}\) (accept \(20^\circ \)) A1
[5 marks]
Question
The diagram shows a tangent, (TP) , to the circle with centre O and radius r . The size of \({\rm{P\hat OA}}\) is \(\theta \) radians.
Find the area of triangle AOP in terms of r and \(\theta \) .[1]
Find the area of triangle POT in terms of r and \(\theta \) .[2]
Using your results from part (a) and part (b), show that \(\sin \theta < \theta < \tan \theta \) .[2]
▶️Answer/Explanation
Markscheme
area of \({\text{AOP}} = \frac{1}{2}{r^2}\sin \theta \) A1
[1 mark]
\({\text{TP}} = r\tan \theta \) (M1)
area of POT \( = \frac{1}{2}r(r\tan \theta )\)
\( = \frac{1}{2}{r^2}\tan \theta \) A1
[2 marks]
area of sector OAP \( = \frac{1}{2}{r^2}\theta \) A1
area of triangle OAP < area of sector OAP < area of triangle POT R1
\(\frac{1}{2}{r^2}\sin \theta < \frac{1}{2}{r^2}\theta < \frac{1}{2}{r^2}\tan \theta \)
\(\sin \theta < \theta < \tan \theta \) AG
[2 marks]