-1
$\begingroup$

I would expect $$cos\left(\frac{3\pi}{4}\right) = cos\left(\frac{\pi}{4}\right)$$

As $3\pi$ puts you on the same spot as $\pi$ does on a goniometric circle. And as the goniometric properties for the $cosine$ describe:

$$cos\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} $$

However it seems to equal $-\frac{\sqrt{2}}{2}$. I don't see where the minus sign comes from.. Any help would be very much appreciated.

  • 1
    $3\pi$ is the same point as $\pi$, but $3\pi/4$ is not the same point as $\pi/4$. – 2017-01-17
  • 0
    The cosine of obtuse angles is negative; you're mistaking with the sine. – 2017-01-17

4 Answers 4

3

In $\frac{3\pi}{4}, \cos(x)$ is negative

enter image description here

  • 0
    This image is very helpful, thanks! I'm assuming that a better way for me to go at it is to literally do $3*180° = 540°$ and to then divide that by 4 which gives 135 and seems to be conform to the position in the image? – 2017-01-17
  • 0
    Purely to know the sign of the value that is. – 2017-01-17
  • 0
    In my opinion, this is the better way. – 2017-01-17
2

You cannot argue that two angles are the same (in the sense of "differ by a multiple of $2\pi$), and then divide them and assume they are still the same. For instance, $0$ and $2\pi$ are the same angle but $\cos(0)\neq \cos\left(\frac{2\pi}4\right)$. So, your reasoning really can't work.

One could note that the identity $\cos(\pi-x)=-\cos(x)$ gives you the answer by noting that $\cos(3\pi/4)=-\cos(\pi/4)=\frac{-\sqrt{2}}2$. More generally, it helps to just know the sign that you should be getting: For angles near $0$, cosine is positive. In particular, cosine is positive on the interval $(-\pi/2,\pi/2)$ and negative in $(\pi/2,3\pi/2)$.

2

The minus sign comes from the fact that cosine represents the $x$-coordinate of the right angle triangle made with the given angle in the unit circle and that $\frac{3 \pi}{4}$ is in the second quadrant, which means that the $x$-coordinate has to be negative.

1

You have $\cos(\frac{3\pi}{4})= \cos(\frac{3\pi}{4} - 2\pi) \not = \cos(\frac{\pi}{4})$. The periodic term may only be added in the argument, not multiplied.