Mathematics 10 — Exercise 7.1 · Q3 (viii)

Full question

Without a calculator evaluate sec 135° exactly.

Step-by-step solution

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

   $$180-135=45$$

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

   $$\sec \left(\mathrm{135°}\right)=-\sec \left(\mathrm{45°}\right)$$

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

   $$\sec \left(\mathrm{45°}\right)=\frac{1}{\cos \left(\mathrm{45°}\right)}$$

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

   $$\sec \left(\mathrm{135°}\right)=\left(-1\right)\times \sqrt{2}$$

Answer

The final answer is shown below.