Question

Sin (x+pi/4) = root2 cos x for 0

Answer 1

I'll assume the rest of the questions is something like for 0 ≤ x < 2π

Solve for x

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

Using the sum angle formula for sine,

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

Of course the sine and cosine of π/4 are the same, both √2/2.

$\dfrac{\sqrt{2}}{2}\sin x+\dfrac{\sqrt{2}}{2} \cos x = \sqrt{2} \cos x$

The square roots of two cancel; let's multiply through by two.

$\sin x + \cos x = 2 \cos x$

$\sin x = \cos x$

$\dfrac{\sin x }{\cos x} = 1$

$\tan x = 1$

$x = \arctan 1$

$x = \frac \pi 4 + \pi k, \quad \textrm{integer } k$

k=0 and k=1 are in the range

Answer: x = π/4 and x = 5π/4

Let's check x=5π/4.

$\sin(\frac{5\pi}{4} + \frac \pi 4) = \sin(\frac{3\pi}{2}) = -1$

$\sqrt{2} \cos(\frac{5\pi}{4})=\sqrt{2} (-\frac{\sqrt{2}}{2}) = -1 \quad\checkmark$