# Find the sign of the expression $\sin \theta+\cos \theta$ when $\theta=100^{\large\circ}$

## 1 Answer

Toolbox:
• $a\cos\theta+b\sin \theta$ are in the form $k\cos \phi$ or $k\sin\phi$
• $k=\sqrt{a^2+b^2}$
$\cos \theta+\sin \theta =\sqrt 2(\large\frac{1}{\sqrt 2}$$\cos \theta+\large\frac{1}{\sqrt 2}$$\sin \theta)$
Multiplying and dividing by $\sqrt{1^2+1^2}$ (i.e) by $\sqrt 2$
$\Rightarrow \sqrt 2(\cos\theta\cos\large\frac{\pi}{4}$$+\sin \theta\sin \large\frac{\pi}{4}) \Rightarrow \sqrt 2\cos(\theta-\large\frac{\pi}{4})=$$\sqrt 2\cos(100^{\large\circ}-45^{\large\circ})$
$\Rightarrow \sqrt 2\cos 55^{\large\circ}$
$\Rightarrow$ a positive quantity $(\cos 55^{\large\circ} > 0)$
answered May 21, 2014

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