How to know when to put calculator in radian or degree mode?

For a take home test that I have, I have some questions that say

Find the exact function value, if it exists.

1. $\sin\left(60^\circ\right)$
2. $\tan\left(-45^\circ\right)$
3. $\cos (5\pi/2)$
4. $\sec (3\pi)$

The second part says

Find the function values. Round to four decimals places.¨:

1. $\cos \left(111.4^\circ\right)$
2. $\sin \left(-18^\circ\right)$
3. $\sec (9π/10)$
4. $\tan (42.5)$

How do I know when my calculator should be in radian or degree mode? Hopefully this makes sense...

• If there is a degree symbol, you should have your calculator in degree mode. Mar 22 '17 at 14:57
• If the input is in degrees (the circle symbol), use degree mode, otherwise use radian mode. Mar 22 '17 at 14:57
• Do you know the difference between degrees and radians? It seems to me that there might be more fundamental problem than typing a value into calculator. Mar 22 '17 at 14:58
• @uniquesolution, that is rather ill-advised. For example, $\tan 42.5$ is not in degrees, while $\tan \pi^\circ$ is. Mar 22 '17 at 15:00
• @uniquesolution, I don't understand what you are disagreeing with. Are you saying that $42.5$ is in degrees, while $\pi^\circ$ is in radians? Mar 22 '17 at 15:03

If there is a degree symbol, ${ \ }^\circ$, then use degree mode.

If there is no degree symbol, then use radian mode. Even if there is no $\pi$ in the number.

In your examples, assuming there are no typos:

1. Sin 60 ° degree mode because there is a degree symbol

2. Tan(-45 °) degree mode because there is a degree symbol

1. Cos 5π/2 radian mode because there is no degree symbol

2. Sec 3π radian mode because there is no degree symbol

I should mention that for those first 4 problems, I think the point is actually not to use a calculator.

The second part says ¨Find the function values. Round to four decimals places.¨:

1. Cos 111.4° degree mode because there is a degree symbol

2. Sin(-18°) degree mode because there is a degree symbol

3. Sec 9π/10 radian mode because there is no degree symbol

4. tan 42.5 radian mode because there is no degree symbol

The ° symbol means "degrees." Any answer marked with that is definitely in degrees.

The technically correct thing to do is to assume that everything is in radians unless otherwise specified. However, humans tend to be bad at being technically correct, so if you haven't been told to use radians unless otherwise specified I would consider making contextual judgement calls. If you have been taught the technically correct rule, definitely use it. This interpretation agrees with the rules of thumb that I am about to give everywhere that it's applicable, leaving the last problem. I would guess that $42.5$ is supposed to be in radians, because everywhere else in the problem the professor has been careful to use the degree symbol, making me think its omission is deliberate. If none of the problems had been marked with a degree symbol, I might think otherwise since $42.5$ is much bigger than $2\pi$.

In contexts where you think your professor has simplified by opting to not use the degree symbol, some general rules of thumb can be applied. If it's unspecified and a $\pi$ shows up, you should assume radians. If it's unspecified and of the form $360/n$ for some integer $n$, use degrees. Another good rule of thumb is that if one interpretation gives an algebraic answer, use that interpretation. I'm unsure if that solves the last problem, though plugging it into Wolfram Alpha will tell you.