Question

find range of sinx + cosx ?​

Answer 1

Answer:

Let f(x)=|sinx|+|cosx|.

Domain : Domain of the function f(x) will be the intersection of domains of sinx and cosx. ...

Therefore, the domain is (-∞,+∞).

Range : Range of any continuous funtion lies inbetween the minimum and maximum value of that function.

Thus the range of the function f(x) is [-√2,√2].

thankyou.

Answer 2

$sin(x) + cos(x)$

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

$= \sqrt{2} ( \cos( \frac{\pi}{4} ) \sin(x) + \sin( \frac{\pi}{4} ) \cos(x) )$

$= \sqrt{2} \sin(x + \frac{\pi}{4} )$

As, For All real values of θ implies

$- 1 \leq \sin( \theta) \leq + 1$

$\implies \sqrt{2} \sin(x + \frac{\pi}{4} ) \in [ - \sqrt{2}, \sqrt{2} ]$

Therefore,

$\boxed{- \sqrt{2} \le\sin(x) + \cos(x) \le + \sqrt{2} , \: \forall \: x \in \mathbb{R}}$