$\cos(\pi/6 *2\pi)$ in radians isn't equal to $\cos(30*360)$ in degrees why?

Question

I put $\cos(\pi/6*2\pi)$ in radians mode on my calculator and i get $-0.9890273166$ and then I put $\cos(30*360)$ in degrees mode and get $1$, the answer is the complete opposite of what it should be and in case if you don't know $\pi/6*2\pi$ converted to degrees equals $30*360$, so I really need an explanation. Thanks.

Answer 1

The formula for Radian-Degree conversion is$${\theta \text{ in Radians}\over \pi}={\theta \text{ in Degrees}\over 180}$$so you may write$$\theta \text{ in Degrees}={180\over \pi}\theta \text{ in Radians}$$so we have$$2\pi\cdot {\pi\over 6}\text{ Rad}=60\pi\text{ Deg}\approx 188.495559 \text{ Deg}$$

Answer 2

Note that $\cos$ is periodic with period $360^\circ$. Therefore, yes, $\cos(30\times360^\circ)=\cos(0)=1$.

And if you use your calculator in radians mode, you will get that $\cos(30\times2\pi)=1$, too. But $\frac\pi6\approx0.523599$, and therefore there's no reason why $\cos\left(\frac\pi6\times2\pi\right)$ should be $1$ too. You seem to think that $\frac\pi6=30$, but that is just wrong. What happens is that $\frac\pi6$ radians is equal to $30^\circ$, but this is an equality between angles, not between numbers.

Answer 3

You can't convert degee to radian each term separately, combine them together and then convert into radians, by multiplying with $ \frac{ \pi}{180} $

Your ( $ \frac{ \pi}{6} \cdot 2 \pi $) is not equal to 30 $ \cdot$360 in degrees.
To convert 30 $ \cdot$360, i.e. 10800 degrees to radians, you need to multiply it by $ \frac{ \pi}{180} $
So, it is equal to 60$ \pi $ radians.

Put cos(60$ \pi$) in radians in your calculator, you will get 1.