An observer views a rocket take off
from a distance of three miles from the launch pad
and tracks the elevation.
Notice how we have three parts here.
For part A, we want to express the height of the rocket
as a function of the angle of elevation, theta.
To do this, let's use the model given here below
to label the given information.
If this is the launch pad, and the observer views
the take off from a distance of three miles away,
the length of this side here would be three miles.
And therefore the length of this side of a right triangle
would be the height of the rocket, which we'll label H.
And therefore the angle of elevation would be
this angle here of a right triangle.
Notice in relation to this angle theta,
the angle of elevation, we'll be working with
the opposite side as well as the adjacent side
of a right triangle.
Therefore if we want an equation that relates
this angle and these two sides,
this should remind us of the tangent function,
where tangent theta is equal to the ratio
of the opposite side to the adjacent side.
So to get started, let's say tangent theta
equals, again, the ratio of the opposite side
to the adjacent side, or H divided by three.
But our goal is to express the height H
as a function of our angle theta,
so now let's solve this for H
by multiplying both sides by three.
This is really three over one, so notice how the threes
simplify to one, giving us the equation
H equals three tangent theta.
So now we do have H as a function of theta,
but we can also write this using function notation
as H of theta equals three tangent theta.
Of course these two equations are equivalent.
This last one is just written using function notation.
Now let's take a look at part B.
We want to express the angle of elevation theta
as a function of the height H of the rocket.
So we'll begin with the equation tangent theta
equals H divided by three again,
but now because we want theta as a function of H,
we'll have to solve this equation for theta, not H.
Since we have tangent theta, if we take the inverse tangent
or arc tangent of both sides of the equation,
we can solve this for theta.
So if we take inverse tangent
of both sides of the equation
we'd have inverse tangent of tangent theta
equals inverse tangent of H divided by three.
Looking at the left side, inverse tangent of tangent theta
equals theta.
And on the right side, we have inverse tangent
of H divided by three.
Now we have theta as a function of H,
so we could write theta of H
equals inverse tangent of H divided by three.
But in our online homework, they prefer us to use
arc tangent rather than inverse tangent,
so this would be equivalent to theta of H
equals arc tangent of H divided by three.
Now let's take a look at part C.
When the angle of elevation is 32 degrees,
we want to find the height of the rocket.
So because we're looking for the height
we'll use the first function that we found
that said H of theta is equal to three tangent theta.
So in this case we have H of 32 degrees
which is equal to three times tangent 32 degrees.
This is not a nice reference angle,
so we'll have to use the calculator
for decimal approximation for the height.
But the first thing we should do
is make sure that we are in degree mode.
So we'll press the mode key,
arrow down to the third row and highlight Degree,
press Enter, go back to the home screen
by pressing Second, Mode for quit,
and now we'll enter three tangent 32,
close parenthesis, enter.
So the distance would be approximately 1.8746,
and the units would be miles.
I hope you found this helpful.