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
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.