Question Video: Using Both Radians and Degrees to Solve Problems Involving Angles Mathematics • First Year of Secondary School

Video Transcript

Find the value of two angles in degrees given their sum is 74 degrees and their difference is 𝜋 over six radians. Give your answer to the nearest degree.

In this question, we’re told that there are two angles. We’re also told that their sum is 74 degrees and their difference is 𝜋 over six radians. When we’re answering a question like this, we’ll have to employ different mathematical skills. We’ll need to use a little bit of algebra to solve this problem. And we’ll also need to know how to convert between angles in degrees and angles in radians.

Let’s begin by saying that we can say that our two angles are called 𝑥 and 𝑦. As we’re told that their sum is 74 degrees, we can say that 𝑥 plus 𝑦 is equal to 74 degrees. Next, we’re told that their difference is 𝜋 over six radians. Remember that difference means subtract, so we can write that 𝑥 subtract 𝑦 is equal to 𝜋 over six radians.

Now that we have two equations with two unknowns, we could solve these. However, the problem is that one of these measurements is in degrees and one of the measurements is in radians. We can either find both of these angles in degrees or both in radians. But if we have a look at the question, we need to give our final answer in degrees, so it would make sense to make sure that they’re both in degrees.

Let’s take this angle then of 𝜋 over six radians and write it as a value in degrees. In order to do this, we need to remember an important conversion between radians and degrees. 𝜋 radians is equal to 180 degrees. Some people prefer to remember that two 𝜋 radians is equal to 360 degrees. But either one will allow us to convert these angles. So, if we take the fact that 𝜋 radians is equal to 180 degrees and the value that we have of 𝜋 over six radians is six times smaller, then that means that our angle in degrees must also be six times smaller than 180 degrees, which means that it must be 30 degrees.

Now that we know that this value of 𝜋 over six radians is actually 30 degrees, we can say that 𝑥 minus 𝑦 is equal to 30 degrees. We can now solve this system of equations by either substitution or by elimination. If we choose to use an elimination method and we wanted to eliminate the 𝑦-variable, then we could add together the first equation and the second equation. Adding the two 𝑥-values would give us two 𝑥. 𝑦 subtract 𝑦 would give us zero. And 74 degrees plus 30 degrees would give us 104 degrees.

We can then find the value of 𝑥 by dividing both sides of this equation by two. So, 𝑥 is equal to 52 degrees. We then take this value of 𝑥 and plug it into either the first equation or the second equation. Using the first equation, then, with 𝑥 is equal to 52 degrees, we’d have that 52 degrees plus 𝑦 is equal to 74 degrees. Subtracting 52 degrees from both sides would give us that 𝑦 is equal to 22 degrees. We can, therefore, give our answer that the two angles must be 52 degrees and 22 degrees. And as they’re already whole-value answers, then we don’t need to worry about rounding to the nearest degree.

It is, of course, always worthwhile checking that our answer is correct. When we were solving it, we used this equation 𝑥 plus 𝑦 equals 74 degrees, so let’s check that if we subtract our angles, we would get 30 degrees. And if we have 52 degrees subtract 22 degrees, we would indeed get 30 degrees, confirming that our two angles are 52 degrees and 22 degrees.