Principal values of inverse trigonometric functions

I have watched multiple times on the videos on inverse cosine and tangent but I couldn't find a relation to these questions below:

Can anyone explain further?

• Mathematical formulae look better in $\LaTeX$. Here is a quick tutorial. – Τίμων Apr 9 '16 at 8:15
• Are you asking for exact values or decimal approximations? Or are you asking for an explanation of those values for negative angles? – Rory Daulton Apr 9 '16 at 8:25
• @RoryDaulton I have edited the questions to the ones similar to the quiz I am working on. The answers available doesnt seems to match the inverse of tangent which I have calculated. – ilovetolearn Apr 9 '16 at 8:28
• @Timon thanks for the edit. – ilovetolearn Apr 9 '16 at 8:28

If you have a scientific calculator with you, the answers are easy: just enter those expressions and press or "=". The calculator will give you the principal value of those functions.

If you do not have a calculator, use your knowledge of those functions to make an approximation which will be close enough to choose one of the four offered possibilities.

In the second question, you want the arccosine of a positive number that is less than one. If you visualize the triangles you will see that the resulting angle is between $0°$ and $90°$. In radians this is between $0$ and $\pi/2$. You know that pi is a little larger than three, so that means between $0$ and a little more than $1.5$. Only one of the given possible answers is in that range, so the answer must be $1.25$. And that is what a calculator gives.

In the first question, the number inside the arctangent is positive, so the resulting angle is between $0$ and a little more than $1.5$. The only given possible answer in that range is $1.31$.

Do you need more detail?

• arctan(3.7) is approx 74.8759... how do you translate this to 0 and 1.5? – ilovetolearn Apr 9 '16 at 8:53
• Your calculator must be in degrees mode. Change it to radians mode and try again. Or, you could convert that degree angle to radians by multiplying by pi and dividing by 180. – Rory Daulton Apr 9 '16 at 8:54
• Why do I have to change it to radian mode? – ilovetolearn Apr 9 '16 at 8:55
• Note the very start of each of the two questions: "The following are all angle measures (in radians...)". – Rory Daulton Apr 9 '16 at 8:56