IP AMaths Notes (Upper Sec, Year 3-4): 12) Trigonometry I

Everything in upper-sec trig is built on radians and the primary identities. Memorise the exact values and practise expressing angles in coterminal forms.

Key facts

• Radian definition: arc length \( s = r\theta \).
• Area of sector: \( A = \tfrac{1}{2} r^2 \theta \).
• Identities: \( \sin^2 \theta + \cos^2 \theta = 1 \); \( 1 + \tan^2 \theta = \sec^2 \theta \).
• Co-function: \( \sin\left(\dfrac{\pi}{2} - \theta\right) = \cos \theta \), \( \cos\left(\dfrac{\pi}{2} - \theta\right) = \sin \theta \).

Worked example 1 — Arc and sector

A circle of radius \( \pu{8 cm} \) subtends an angle of \( \dfrac{5\pi}{12} \). Find the arc length and sector area.

1. Arc length: \( s = r\theta = 8 \times \dfrac{5\pi}{12} = \dfrac{10\pi}{3} \pu{cm} \).
2. Area: \( A = \dfrac{1}{2} r^2 \theta = \dfrac{1}{2} \times 64 \times \dfrac{5\pi}{12} = \dfrac{80\pi}{12} = \dfrac{20\pi}{3} \pu{cm2} \).

Worked example 2 — Identity manipulation

Show that \( \tan \theta + \cot \theta \geq 2 \) for \( 0 < \theta < \dfrac{\pi}{2} \).

1. Express \( \tan \theta + \cot \theta = \dfrac{\sin \theta}{\cos \theta} + \dfrac{\cos \theta}{\sin \theta} \).
2. Combine: \( \dfrac{\sin^2 \theta + \cos^2 \theta}{\sin \theta \cos \theta} = \dfrac{1}{\sin \theta \cos \theta} \).
3. Using double-angle identity, \( \sin \theta \cos \theta = \dfrac{1}{2} \sin 2\theta \).
4. Hence \( \tan \theta + \cot \theta = \dfrac{2}{\sin 2\theta} \).
5. For \( 0 < \theta < \dfrac{\pi}{2} \), \( 0 < 2\theta < \pi \), so \( \sin 2\theta \leq 1 \).
6. Therefore expression is \( \geq 2 \) with equality when \( \sin 2\theta = 1 \) i.e. \( \theta = \dfrac{\pi}{4} \).

Try this

Convert \( 210^\circ \) to radians, then evaluate \( \sin 210^\circ \) and \( \cos 210^\circ \) exactly using ASTC rules.