Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $\sqrt{2}$

Answer 1

Answer: (D)

$\sqrt{2}$

Formula used:

$\sin (\alpha +\beta )=\sin \alpha \cos \beta +\cos \alpha \sin \beta$

$\sin \theta +\cos \theta$

$=\sqrt{2}(\frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta )$

$=\sqrt{2}(\cos {45}^{\circ }\sin \theta +\sin {45}^{\circ }\cos \theta )$

$=\sqrt{2}\sin ({45}^{\circ }+\theta )$
and maximum value of $\sin ({45}^{\circ }+\theta )=1$

$\therefore$ maximum value of $\sin \theta +\cos \theta =\sqrt{2}$