Triangles, radians and sectors

Triangles (MM4.1)

• Sine rule: \(\dfrac{a}{\sin A} = \dfrac{b}{\sin B} = \dfrac{c}{\sin C}\). Cosine rule: \(a^2 = b^2 + c^2 - 2bc\cos A\), i.e. \(\cos A = \dfrac{b^2 + c^2 - a^2}{2bc}\).
• Area \(= \frac12 ab\sin C\). Heron-style: find \(\cos\) from the cosine rule, then \(\sin\) from \(\sin^2 + \cos^2 = 1\).
• Ambiguous case: given sides \(a, b\) and angle \(A\) (opposite \(a\)) with \(b\sin A < a < b\), there are two triangles (an acute and an obtuse value of \(B\)). If \(a \geq b\) or \(a = b\sin A\) there is one; if \(a < b\sin A\), none.
• Largest angle is opposite the longest side; \(a^2 + b^2 < c^2\) means the angle opposite \(c\) is obtuse.

Radians, arcs and sectors (MM4.2)

• \(\pi\) radians \(= 180^\circ\); \(30^\circ = \frac\pi6\), \(45^\circ = \frac\pi4\), \(60^\circ = \frac\pi3\), \(90^\circ = \frac\pi2\).
• Arc length \(s = r\theta\); sector area \(\frac12 r^2\theta\); segment area \(\frac12 r^2(\theta - \sin\theta)\) (\(\theta\) in radians).
• Perimeter of a sector \(= 2r + r\theta\).
• Overlapping circles: the common region is two segments; the chord subtends \(2\theta\) at each centre.