Video Transcript
Find the value of two angles in
degrees given their sum is 74 degrees and their difference is ๐ over six
radians. Give your answer to the nearest
degree.
In this question, weโre told that
there are two angles. Weโre also told that their sum is
74 degrees and their difference is ๐ over six radians. When weโre answering a question
like this, weโll have to employ different mathematical skills. Weโll need to use a little bit of
algebra to solve this problem. And weโll also need to know how to
convert between angles in degrees and angles in radians.
Letโs begin by saying that we can
say that our two angles are called ๐ฅ and ๐ฆ. As weโre told that their sum is 74
degrees, we can say that ๐ฅ plus ๐ฆ is equal to 74 degrees. Next, weโre told that their
difference is ๐ over six radians. Remember that difference means
subtract, so we can write that ๐ฅ subtract ๐ฆ is equal to ๐ 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 theyโre both in degrees.
Letโs take this angle then of ๐
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. ๐ radians is equal to 180
degrees. Some people prefer to remember that
two ๐ radians is equal to 360 degrees. But either one will allow us to
convert these angles. So, if we take the fact that ๐
radians is equal to 180 degrees and the value that we have of ๐ 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
๐ over six radians is actually 30 degrees, we can say that ๐ฅ minus ๐ฆ 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 ๐ฆ-variable, then we could add together the
first equation and the second equation. Adding the two ๐ฅ-values would give
us two ๐ฅ. ๐ฆ subtract ๐ฆ would give us
zero. And 74 degrees plus 30 degrees
would give us 104 degrees.
We can then find the value of ๐ฅ by
dividing both sides of this equation by two. So, ๐ฅ is equal to 52 degrees. We then take this value of ๐ฅ and
plug it into either the first equation or the second equation. Using the first equation, then,
with ๐ฅ is equal to 52 degrees, weโd have that 52 degrees plus ๐ฆ is equal to 74
degrees. Subtracting 52 degrees from both
sides would give us that ๐ฆ is equal to 22 degrees. We can, therefore, give our answer
that the two angles must be 52 degrees and 22 degrees. And as theyโre already whole-value
answers, then we donโt 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 ๐ฅ plus ๐ฆ equals 74 degrees, so letโs 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.
