Question Video: Using Both Radians and Degrees to Solve Problems Involving Angles Mathematics

Two angles in a triangle are 55° and 7𝜋/18 radians. Find the value of the third angle giving the answer in radians in terms of 𝜋.

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Video Transcript

Two angles in a triangle are 55 degrees and seven 𝜋 over 18 radians. Find the value of the third angle giving the answer in radians in terms of 𝜋.

In this question, we’re given that there’s a triangle. One of the angles, we’re told, is 55 degrees. And the other angle is seven 𝜋 over 18 radians. We need to find out the value of the third angle in this triangle. And we need to give the answer in radians.

The first problem we should notice here is that one of the angles is given in degrees and the other one is given in radians. We might be a little confused and think that radians are something that only involves circles. But just remember that radians are like degrees; they’re just a unit of measurement for angles. What we want to do in this question is to make sure that all the angles are given in the same unit of measurement. We could change them all to degrees, or we could change them all to radians. But we should notice that the question asks for the angle in radians at the end, so it might be more sensible to change our value in degrees into a value in radians.

When it comes to converting between degrees and radians, there are two common conversions we can remember, either 180 degrees is equal to 𝜋 radians or 360 degrees is equal to two 𝜋 radians. Remembering just one of these will allow us to convert any measurement in degrees into a measurement in radians. So, let’s take this conversion of 180 degrees is equal to 𝜋 radians and use it to change 55 degrees into a value in radians. If we take this interim step of finding one degree, then we can notice that to go from 180 to one, we must divide by 180.

This means that we need to do the same on the other side with our value in radians. And 𝜋 divided by 180 can be written as 𝜋 over 180 radians. To go from one degree to 55 degrees, we must multiply by 55. We can then simplify this value on the right-hand side. So, we can say that 55 degrees is equal to 11𝜋 over 36 radians. We can notice, of course, that there is another way we could’ve solved this. Instead of finding one degree, we might have converted five degrees into radians. To go from 180 degrees to five degrees, we must have divided by 36. Then, to change five degrees into 55 degrees, we would’ve needed to multiply both sides by 11. Either way would have given us the value of 11𝜋 over 36 radians.

Now that we’ve calculated that this angle of 55 degrees is equivalent to 11𝜋 over 36 radians, let’s see if we can find this third angle in the triangle. We’ll need to remember that the angles in a triangle add up to 180 degrees. If we call this unknown angle in the triangle 𝑥, we could begin to write that the three angles must sum to — Oh dear! They can’t add up to 180 degrees; it needs to be a value in radians. But we already know that 180 degrees is equal to 𝜋 radians. Therefore, these three angles must add to 𝜋.

We’ll now need to do some fraction arithmetic. And we’ll remember that when we’re adding fractions, they need to have the same denominator. In order to write our second fraction with the denominator of 36, we’ll need to multiply the numerator and denominator by two. So, our second fraction will be 14𝜋 over 36. When we’re adding fractions and they’re the same denominator, we add the numerators. So, we’ll have 25𝜋 over 36 plus 𝑥 is equal to 𝜋. In order to find 𝑥, we subtract 25𝜋 over 36 from both sides of this equation. In order to help us solve this, it might be helpful to take out a factor of 𝜋. So, we’ll have 𝑥 is equal to 𝜋 multiplied by one minus 25 over 36.

Using the fact that this whole value of one must be 36 over 36, then we get that 𝑥 is equal to 11𝜋 over 36 radians. And that’s our answer for the third angle in this triangle. And it’s given in radians and in terms of 𝜋.

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