Convert negative 105 degrees into radians in terms of 𝜋.
The first thing we should note is that even though this angle in degrees is negative, we can still convert it into radians by following the usual process. We can do this by either recalling that to convert an angle in degrees to one in radians, we multiply the given angle by 𝜋 over 180 degrees. Or we can use the conversion that 180 degrees is equal to 𝜋 radians. Knowing this conversion will allow us to work out the calculation in the statement above. So let’s use this first fact to work out negative 105 degrees in radians. And so we take negative 105 and multiply it by 𝜋 over 180.
Now, since we are told to leave our answer in terms of 𝜋, we can assume that we are not allowed to use a calculator here. This means that we will need to simplify our calculation by using fractions. We will need to write 105 over 180 in its simplest form. We might firstly notice that both of these values are divisible by five, which would leave us with negative 21 over 36𝜋 radians. But of course 21 and 36 are both divisible by three. Dividing both of these values by three would give us negative seven twelfths 𝜋 radians. Because the value of 𝜋 is multiplied by negative seven over 12, then we can write it beside the fraction or as part of the numerator. And so we can give the answer that negative 105 degrees is equivalent to negative seven 𝜋 over 12 radians.