Question

The minimum value of $\cos\theta + \sin\theta$ is

• A. 0
• B. $- \sqrt{2}$
• C. $1/2$
• D. $\sqrt{2}$

Answer 1

Answer: (B)

$- \sqrt{2}$

Answer 2

Let $f(x) = \cos\theta + \sin\theta = \sqrt{2}\cos\left( \theta - \frac{\pi}{4} \right)$

Since $- 1 \leq \cos\left( \theta - \frac{\pi}{4} \right) \leq 1$

⇒ $- \sqrt{2} \leq \sqrt{2}\cos\left( \theta - \frac{\pi}{4} \right) \leq \sqrt{2}$

Thus, the minimum value of $f(x)$ is $- \sqrt{2}$.