Applications of Trigonometry - Mathematics, Class 10 Video Lecture

in today's lesson we're going to take a
look at some applications of
trigonometry suppose we have a building
we have a little old lady down on the
ground and she's looking at the bottom
of the building she decides to look
upward toward the top the angle that's
been formed by her looking up is called
the angle of elevation you can remember
that because when you look up it's
called elevation and if you want to go
up you'll take an elevator now suppose
the little old lady gets inside the
building she takes the elevator to the
top and now she's standing up on the top
of the building she's looking straight
ahead into the horizon she decides to
look down to the ground the angle that's
formed when she looks down is called the
angle of depression you can remember
that because if you have depression
you'll feel down so what does this
really mean well first of all here's an
interesting fact the angle of depression
is always the same is the angle of
elevation so if the angle of depression
is 45 degrees the angle of elevation is
also 45 degrees this is actually very
handy and you'll see how this is as we
look through some examples in example 1
we have a tree John wants to know how
high the tree is but he doesn't really
feel like climbing up to the top and
measuring down he decides to use
trigonometry whenever we use
trigonometry we're talking about
triangles and so we'll draw a triangle
next to the tree he walks 100 feet from
the bottom of the tree and so we'll
begin by labeling our triangle 100 on
the bottom that's the side that's on the
ground then he looks up the angle of
elevation is 33 degrees and so we'll
label that in our triangle also now the
question says how tall is the tree
that's what I want to find out and so
I'll call the side along the tree X
notice that I have here the opposite and
the adjacent which of our three trig
ratios uses the opposite in the adjacent
well that's the tangent
so the tangent of 33 degrees equals
opposite x over adjacent 100 now I can
cross multiply and plug that into my
calculator and I get x equals 65 the
tree must be 65 feet tall in example 3
we have a building that's 50 feet high
we have an observer and he knows that
the angle of elevation looking up from
the ground to the top of the building is
41 degrees the question is how far away
from the base of the building is the
observer well we're going to use
trigonometry and so we'll draw a
triangle right alongside the building
now let's fill in the information that
we have the building is 50 feet high the
angle of elevation is 41 degrees and the
observer wants to know how far we are
from the base of the building now that
we've filled in our triangle we can use
trigonometry to find the length of that
side please pause the video here find
the length of the side marked X and then
come on back and we'll see how you did
we began by looking at the sides that we
have we had our opposite and our
adjacent opposite and adjacent is the
tangent ratio and so we have the tangent
of 41 degrees equals 50 over X now we
cross multiply and solve for x x equals
58 which means the observer must be 58
feet from the base of the building next
let's take a look at example 4 we have
an airplane that's flying at a height of
2 miles above the ground and we know
that if we go down to the ground it's
about five miles from the airplane to
the airport we want to know what the
angle of depression is from the airplane
to the airport for example what angle
does the pilot need to put the plane
down so that it can land at the airport
we draw a triangle and we label what we
have the airplane is 2 miles above the
ground
the distance from the airport to the
airplane along the ground is five miles
we want to know the angle of depression
the angle that the pilots looking down
to see the airport remember the angle of
depression has the same measure is the
angle of elevation that actually makes
sense because if the pilot is looking
down at the airport and the person at
the airport is looking up at the
airplane they must be looking at the
same angle now we can use our
trigonometry we have our opposite and
our adjacent once again so the tangent
of angle X is two over five now we find
the angle with the inverse and we find
out that the angle of elevation is
twenty-one point eight degrees which
means the angle of depression is also
twenty-one point eight degrees our final
example example seven is for you to try
we have a 40-foot ladder it's leaning
against a 12-foot tall building they
want to know the angle of elevation that
the latter makes with the ground please
label the triangle with the sides and
the missing angle and then come back and
we'll take a look at your work we began
by drawing a triangle the ladder is the
hypotenuse and the left-hand side of the
triangle is the building we know that
the building is 12 feet tall the ladder
is 40 and the angle of elevation is X
that's what we want to find now let's
find the measure of the angle please
pause the video here find the measure of
angle X and then come on back let's see
how you did
we have the opposite and the hypotenuse
opposite and hypotenuse is our sine
ratio so the sine of angle X is 12 over
40 we use the inverse to find the
measure of the angle we get that the
angle is approximately 17 degrees
now to follow up with this the ladder
manufacturer has issued a statement in
the owners manual that statement says
for safety a rigid ladder should be
leaned at an angle of about 15 degrees
to the vertical at steeper angles the
ladder is at risk of toppling backwards
when the climber leans away from it at
shallower angles the ladder may lose its
grip on the ground and fall is our
ladder in compliance well it says that
we should have an angle of no more than
15 degrees our angle happens to be 17
degrees which is a problem and therefore
we're not in compliance we run the risk
of toppling over so here's everything
you need to know about applications of
trigonometry when you have it draw a
picture draw a picture of the tree draw
a picture of the building draw a picture
of the lamppost or whatever you are
working with draw your triangle next to
it
label the sides and then use your trig
ratios to find the piece of information
you want that's everything you need to
know about working with basic
applications of trigonometry