Mathematics 10 — Exercise 7.1 · Q3 (iv)

Full question

Without a calculator evaluate cosec 120° exactly.

Step-by-step solution

1. The angle lies in quadrant II. Subtract it from 180° to find the reference angle.

   $$180-120=60$$

2. Use the supplementary-angle identity and its quadrant sign.

   $$\mathrm{cosec}\left(\mathrm{120°}\right)=\mathrm{cosec}\left(\mathrm{60°}\right)$$

3. Write this ratio as the reciprocal of its basic ratio.

   $$\mathrm{cosec}\left(\mathrm{60°}\right)=\frac{1}{\sin \left(\mathrm{60°}\right)}$$

4. Insert the exact 30°, 45° or 60° value and simplify the fraction or root.

   $$\mathrm{cosec}\left(\mathrm{120°}\right)=\frac{2}{3}\times \sqrt{3}$$

Answer

The final answer is shown below.