Linear Trig Equations: Solving sin(x + pi/4) = √2 cos x

[Veronica_Oles]

Homework Statement

Solve sin (x + pi/4) = √2 cos x

Homework Equations

The Attempt at a Solution

sinx*cos(pi/4) + cosx*sin(pi/4) = √2 cos x
√2/2 sinx + √2/2 cosx = √2 cos x
not sure if I am on the right track? or where would I go from here? would I bring √2 cos x to the left side?

 

[LCKurtz]

Homework Statement

Solve sin (x + pi/4) = √2 cos x

Homework Equations

The Attempt at a Solution

sinx*cos(pi/4) + cosx*sin(pi/4) = √2 cos x
√2/2 sinx + √2/2 cosx = √2 cos x
not sure if I am on the right track? or where would I go from here? would I bring √2 cos x to the left side?

I would note that $\frac {\sqrt 2} 2 = \frac 1 {\sqrt 2}$ and multiply both sides by $\sqrt 2$.

 

[Veronica_Oles]

I would note that $\frac {\sqrt 2} 2 = \frac 1 {\sqrt 2}$ and multiply both sides by $\sqrt 2$.

Did that now I'm left with (sinx + cosx) = 2cosx, I'm stuck now? Tried bringing to other side and does not work and tried cancelling out the cosx but that does not work.

 

[LCKurtz]

Did that now I'm left with (sinx + cosx) = 2cosx, I'm stuck now? Tried bringing to other side and does not work and tried cancelling out the cosx but that does not work.

Show us what you get when you simplify it. Telling us it didn't work doesn't help us help you when we don't know what you did.

 

[lurflurf]

just use the identity
$$\sin\left(x+\frac{\pi}{4}\right)=\sin\left(x-\frac{\pi}{4}\right)+\sqrt{2}\cos(x)$$
or equivalently
$$\sin\left(x+\frac{\pi}{4}\right)-\sin\left(x-\frac{\pi}{4}\right)=\sqrt{2}\cos(x)$$

 

[Veronica_Oles]

Show us what you get when you simplify it. Telling us it didn't work doesn't help us help you when we don't know what you did.

Lol I got the answer.

(sinx + cosx)/cosx = (2cosx)/cosx

Now I am left with

(sinx/cosx) + 1 = 2

sinx/cosx = 2-1

tanx = 1

x = tan^-1(1)

x = pi/4

or

x = pi + pi/4 = 5pi/4

 

[LCKurtz]

How about angles coterminal with those?