Year 12 (9709) — Circular Measure Review
Use radians for all circular-measure formulas unless stated otherwise.
1) Radian recap
Full circle: \(2\pi\) rad, half: \(\pi\) rad, quarter: \(\tfrac{\pi}{2}\) rad.
Conversion: \( \;x^\circ = x\cdot \dfrac{\pi}{180}\text{ rad}\), \(\;\; y\text{ rad} = y\cdot \dfrac{180}{\pi}^\circ\).
Example — Convert \( 150^\circ \) to radians
\[
150^\circ = 150\cdot \frac{\pi}{180}=\frac{5\pi}{6}\text{ rad}.
\]
2) Arc length
For a circle of radius \(r\) with central angle \(\theta\) (in radians):
\[
\text{Arc length } \ell = r\theta.
\]
Example — Find arc length for \(r=7\text{ cm}\), \(\theta=\tfrac{2\pi}{3}\)
\[
\ell = r\theta = 7\cdot \frac{2\pi}{3}=\frac{14\pi}{3}\text{ cm}\;\;(\approx 14.66\text{ cm}).
\]
3) Sector: area and perimeter
Area of a sector:
\[
A_{\text{sector}}=\frac{1}{2}r^{2}\theta.
\]
Perimeter of a sector (two radii + arc):
\[
P_{\text{sector}}=2r+r\theta.
\]
Example — \(r=5\text{ cm}\), \(\theta=1.2\text{ rad}\)
\[
A=\tfrac12\cdot 5^{2}\cdot 1.2=15\text{ cm}^2,\qquad
P=2\cdot 5 + 5\cdot 1.2 = 10+6=16\text{ cm}.
\]
4) Segment (minor): area and perimeter
Segment area (sector minus isosceles triangle):
\[
A_{\text{seg}}=\frac{1}{2}r^{2}\,\big(\theta-\sin\theta\big).
\]
Chord length for angle \(\theta\):
\[
c=2r\sin\!\left(\frac{\theta}{2}\right).
\]
Segment perimeter (arc + chord):
\[
P_{\text{seg}}=r\theta + 2r\sin\!\left(\frac{\theta}{2}\right).
\]
Example — \(r=9\text{ cm}\), \(\theta=1.4\text{ rad}\)
\[
A_{\text{seg}}=\tfrac12\cdot 9^{2}\,(1.4-\sin 1.4)
=40.5\,(1.4-0.9854)
\approx 40.5\cdot 0.4146
\approx 16.8\text{ cm}^2.
\]
\[
c=2\cdot 9 \sin(0.7)\approx 18\cdot 0.645=11.61\text{ cm}.
\]
\[
P_{\text{seg}}=r\theta+c=9\cdot 1.4 + 11.61 \approx 12.6+11.61=24.21\text{ cm}.
\]
5) Right triangles — SOHCAHTOA
For an acute angle \(\theta\) in a right triangle:
\[
\sin\theta=\frac{\text{opp}}{\text{hyp}},\quad
\cos\theta=\frac{\text{adj}}{\text{hyp}},\quad
\tan\theta=\frac{\text{opp}}{\text{adj}}.
\]
Example — hyp = 13, adj = 5, find \(\cos\theta\) and \(\sin\theta\)
\[
\cos\theta=\frac{5}{13},\qquad
\sin\theta=\sqrt{1-\cos^{2}\theta}=\sqrt{1-\frac{25}{169}}=\frac{12}{13}.
\]
6) Sine Rule
For any triangle \(ABC\) with sides \(a,b,c\) opposite angles \(A,B,C\):
\[
\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}.
\]
Use when you have a pair (side with its opposite angle).
Example — Given \(A=40^\circ\), \(B=65^\circ\), \(a=7.2\text{ cm}\), find \(b\)
First \(C=180^\circ-40^\circ-65^\circ=75^\circ\). Then
\[
\frac{a}{\sin A}=\frac{b}{\sin B}
\Rightarrow b=\frac{\sin 65^\circ}{\sin 40^\circ}\cdot 7.2
\approx \frac{0.9063}{0.6428}\cdot 7.2
\approx 10.16\text{ cm}.
\]
7) Cosine Rule
For side \(a\) opposite angle \(A\):
\[
a^{2}=b^{2}+c^{2}-2bc\cos A
\]
(and cyclic permutations). Use for SAS or SSS problems.
Example — \(b=8\), \(c=11\), included angle \(A=52^\circ\). Find \(a\)
\[
a^{2}=8^{2}+11^{2}-2\cdot 8\cdot 11\cos 52^\circ
=64+121-176\cos 52^\circ.
\]
With \(\cos 52^\circ\approx 0.6157\):
\[
a^{2}\approx 185-176(0.6157)=185-108.36=76.64
\Rightarrow a\approx 8.76.
\]
8) Area of a triangle
If two sides \(a,b\) and the included angle \(C\) are known:
\[
\text{Area}=\frac{1}{2}ab\sin C.
\]
Example — \(a=9\), \(b=12\), \(C=35^\circ\)
\[
\text{Area}=\tfrac12\cdot 9\cdot 12\cdot \sin 35^\circ
=54\cdot 0.574\approx 31.0\text{ (square units)}.
\]
9) Mixed application (arc, chord, triangle area)
In a circle of radius \(r\), a chord subtends \(\theta\) at the centre.
The isosceles triangle formed by two radii has:
\[
\text{base } c=2r\sin(\tfrac{\theta}{2}),\qquad
\text{area }=\tfrac12 r^{2}\sin\theta.
\]
The corresponding sector has \(A_{\text{sector}}=\tfrac12 r^{2}\theta\) and arc length \(r\theta\).
Example — \(r=10\), \(\theta=1.1\text{ rad}\)
\[
c=2\cdot 10\sin(0.55)\approx 20\cdot 0.5227=10.45,
\]
\[
\text{Area}_{\triangle}=\tfrac12\cdot 10^{2}\sin(1.1)=50\cdot 0.8912\approx 44.56,
\]
\[
\text{Area}_{\text{sector}}=\tfrac12\cdot 10^{2}\cdot 1.1=55,
\;\;\text{so }A_{\text{seg}}=55-44.56\approx 10.44.
\]
\[
\ell_{\text{arc}}=10\cdot 1.1=11,\qquad
P_{\text{seg}}=\ell_{\text{arc}}+c\approx 11+10.45=21.45.
\]
Key Points
- Always use radians for \( \ell=r\theta \), \(A_{\text{sector}}=\tfrac12 r^{2}\theta\), and \(A_{\text{seg}}=\tfrac12 r^{2}(\theta-\sin\theta)\).
- Segment perimeter = arc + chord, with \(c=2r\sin(\tfrac{\theta}{2})\).
- Sine Rule for pairs (angle & opposite side), Cosine Rule for SAS/SSS.
- Triangle area (non-right-angled): \(\tfrac12 ab\sin C\).<\li>
- Right triangles: SOHCAHTOA.