Write 'True' or 'False' and justify your answer.
The value of $\sin \theta + \cos \theta$ is always greater than $1$.
The value of $\sin \theta + \cos \theta$ is always greater than $1$.
(B) False.
The expression $\sin \theta + \cos \theta$ can be written as $\sqrt{2} (\frac{1}{\sqrt{2}} \sin \theta + \frac{1}{\sqrt{2}} \cos \theta) = \sqrt{2} \sin(\theta + 45^{\circ})$.
For $\theta = 0^{\circ}$,the value is $\sin 0^{\circ} + \cos 0^{\circ} = 0 + 1 = 1$.
Since the value can be equal to $1$ (e.g.,at $\theta = 0^{\circ}$ or $\theta = 90^{\circ}$),the statement that it is 'always greater than $1$' is false.
The expression $\sin \theta + \cos \theta$ can be written as $\sqrt{2} (\frac{1}{\sqrt{2}} \sin \theta + \frac{1}{\sqrt{2}} \cos \theta) = \sqrt{2} \sin(\theta + 45^{\circ})$.
For $\theta = 0^{\circ}$,the value is $\sin 0^{\circ} + \cos 0^{\circ} = 0 + 1 = 1$.
Since the value can be equal to $1$ (e.g.,at $\theta = 0^{\circ}$ or $\theta = 90^{\circ}$),the statement that it is 'always greater than $1$' is false.
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