1. Tardigrade
  2. Question
  3. Mathematics
  4. The maximum value of 1+ sin ((π/4)+θ)+2 cos ((π/4)-θ) for all real values of θ is

Question Error Report

Thank you for reporting, we will resolve it shortly

Back to Question

Q. The maximum value of $1+\sin \left(\frac{\pi}{4}+\theta\right)+2 \cos \left(\frac{\pi}{4}-\theta\right)$ for all real values of $\theta$ is

Trigonometric Functions

Solution:

$\Rightarrow 1+\frac{1}{\sqrt{2}} \sin \theta+\frac{1}{\sqrt{2}} \cos \theta+2\left(\frac{1}{\sqrt{2}} \cos \theta+\frac{1}{\sqrt{2}} \sin \theta\right)$
$\Rightarrow 1+\left(\frac{1}{\sqrt{2}}+\sqrt{2}\right) \sin \theta+\left(\frac{1}{\sqrt{2}}+\sqrt{2}\right) \cos \theta$
$\Rightarrow$ Maximum value is
$\sqrt{\left(\frac{3}{\sqrt{2}}\right)^{2}+\left(\frac{3}{\sqrt{2}}\right)^{2}}+1=\sqrt{\frac{9}{2}+\frac{9}{2}}+1=4$