# Why calculate cos on different calculator get different results?

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 another calcuator: Could anyone explain me why this happens and which one is correct?

It is very confusing for me.

Thanks for any responds.

• Did you mean $35$ degrees or $35$ radians?
– lulu
Jan 11, 2022 at 15:58
• @lulu I think as you Jan 11, 2022 at 15:59
• @lulu, I am calculating degrees Jan 11, 2022 at 16:00
• Do your calculators know that? $35$ radians is $2005$ degrees.
– lulu
Jan 11, 2022 at 16:01
• You may be trying to calculate degrees but the second calculator is calculator is calculator radians. $\cos 35^0 = 0.8195204$ but $\cos 35\ radians = -0.90369255$. That's what's happening. Jan 11, 2022 at 16:05

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.

• Thanks so much for the detailed answer! Jan 11, 2022 at 21:08