# Does sin180+sin45=sin225?

Nov 25, 2016

Ehm...I do not think so...

#### Explanation:

I imagine the arguments in degrees so that if you plot your sin function you get:

As you can see:
$\sin \left({180}^{\circ}\right) = 0$
while $\sin \left({45}^{\circ}\right)$ is positive and $\sin \left({225}^{\circ}\right)$ is negative.

I also considered the possibility to have them in radians but it doesn't work either...

Nov 25, 2016

No

#### Explanation:

Remember that taking the $\sin$ of something is a function that is unique only to that number (let's imagine that we're in a range of $0 < \theta < 2 \pi$).

So the $\sin 180$ is a certain value and the $\sin 45$ is a certain value. The $\sin 225$ is also a separate certain value. You cannot find the $\sin$ of two different values, and add them up to be the $\sin$ of their sum.

Think of it this way:

$\sqrt{4} + \sqrt{25} = \sqrt{29}$
$2 + 5 = \sqrt{29}$
$7 \cancel{=} \sqrt{29}$

In the same idea, $\sin$ does not work that way.

Like Gio explained, the graphs are also different.

But how can you trust just plain words? Let's actually work out this problem.

$\sin 180$ = 0

$\sin \left(45\right) = \frac{\sqrt{2}}{2}$

$\sin 225 = - \frac{\sqrt{2}}{2}$

So:

$\sin 180 + \sin 45 = \sin 225$

$0 + \frac{\sqrt{2}}{2} \cancel{=} - \frac{\sqrt{2}}{2}$

And that's why you cant work with $\sin$ like that!