I just recently start to learn cos. I found out when I use different calculators to calculate cos, sometimes may get different results which is quite confusing.
For example, cos(35), I am calculating degrees.
on 1 calculator:
on another calcuator:
Could anyone explain me why this happens and which one is correct?
It is very confusing for me.
Thanks for any responds.
On the first calculator the calculator is calibrated to take input in degrees whereas in second it is in radians .
The first calculator is doing degrees. $\cos 35^\circ \approx 0.819$. This is reasonable because $0< 35 < 90$ so the $\sin/\cos$ values will both be positive. $30 < 35 < 45$ so $\cos 30^\circ =\frac {\sqrt 3}2 > \cos 35^\circ > \cos 45^\circ =\frac 1{\sqrt 2}$ so the value $0.819...$ is reasonable. $-0.9036...$ is not.
The second calculator is doing radians and for radian input $35$ is a strange input but not impossible. $35 = \frac {35}{\pi} \pi \approx 11.14 \pi$. As trig values repeat every $2\pi$ turns, $\cos 35 = \cos 11.14 \pi = \cos 1.14\pi$. As $\cos \pi = -1$, we have $\cos 1.14$ will be "kind of" close to $-1$. And $-0.9036...$ is reasonable for radians.
Alternatively. $360^\circ = 2\pi \ radians$ so $1\ radian = \frac {360}{2\pi}^\circ$ and $35\ radians = 35\times \frac {360}{2\pi}^\circ \approx 2005.35^\circ$. As trig values repeat every $360^\circ$, we know $\cos 2005.35^\circ = \cos (5\cdot 360 + 205.35)^\circ = \cos 205.35^\circ = \cos (180^\circ + 25.35^\circ)$. As $\cos 180 = -1$ then $\cos 205.35$ should be "kind of" close to $-1$ and that's a reasonable answer for $\cos 2005.35^\circ$.
(More guestimation: $\cos (180 + k)^\circ = - \cos k^\circ$ so $\cos 205^\circ = \cos (180 + 25)^\circ = -\cos 25^\circ$. As $0 < 25 < 30$ we have $1=\cos 0 > \cos 25 > \cos 30 = \frac {\sqrt 3}2 \approx 0.866$ so $\cos 25^\circ \approx 0.9036...$ is a reasonable answer and $-0.9306...$ is a reasonable answer for $\cos 2005^\circ$.)
tl;dr: Doing trig with degrees (1 circle = $360^\circ$) and doing trig with radians (1 circle = $2\pi$ radians) will have very different results. One calculator is using degrees. The other is using radians.... That's my final answer.
