Video Transcript
A plane travels 800 meters along
the runway before taking off at an angle of 10 degrees. It travels a further 1,000 meters
at this angle as seen in the figure. Work out the distance of the plane
from its starting point. Give your answer to two decimal
places.
Looking at the diagram, we can see
that we have a triangle. We want to calculate the distance
of the plane from its starting point. Thatโs this length here, which we
can refer to as ๐ meters. We know the lengths of the other
two sides in this triangle. They are 800 meters and 1,000
meters. And using the fact that angles on a
straight line sum to 180 degrees, we can work out the size of this angle here. Itโs 180 degrees minus 10 degrees,
which is 170 degrees.
As this is a non-right-angled
triangle, we need to answer this problem using either the law of sines or the law of
cosines. So the first step is to decide
which of these we need. And that will depend on the
specific combination of information weโve been given and what we want to
calculate.
In this triangle, we know two sides
and the included angle. And we want to calculate the third
side. We recall then that this means we
should be using the law of cosines. Letโs recall the law of
cosines. Itโs ๐ squared equals ๐ squared
plus ๐ squared minus two ๐๐ cos ๐ด. Now, thereโs no need to actually
label our triangle using the letters ๐ด, ๐ต, and ๐ถ. Instead, we just remember that the
lowercase letters ๐ and ๐ represent the two sides we know and the capital letter
๐ด represents the included angle.
So using 800 and 1,000 as the two
side lengths ๐ and ๐ and 170 degrees as the angle ๐ด, we have the equation ๐
squared equals 800 squared plus 1,000 squared minus two times 800 times 1,000 times
cos of 170 degrees. We can either type this directly
into our calculator or it may be a good idea to break the calculation down into some
stages. In either case, we arrive at ๐
squared equals 3,215,692.405.
Now, we must remember that this is
๐ squared. It isnโt ๐, so we arenโt
finished. We have to square root in order to
find the value of ๐. Itโs a really common mistake though
to forget to do this. Square rooting gives ๐ equals
1,793.235178. The question asks us to give our
answer to two decimal places. So rounding appropriately, weโve
worked out the distance of the plane from its starting point. Itโs 1,793.24 meters to two decimal
places.
