A footballer kicks a ball into a
goal post. The angle of elevation between the
trajectory of the ball and the pitch is 38 degrees. The ball hits the top of the goal
post at a height of 2.44 meters. Find the horizontal distance 𝑋
between the footballer and the goal, giving the answer to two decimal places.
So as we see from the question that
this actual scenario has been modelled with a right-angled triangle. So I’ve got this diagram here. The fact that it’s a right-angled
triangle is key because it means we can decide whether to use Pythagorean theorem or
And in this question, we can see
that we’re given an angle and a side length. And we want to find another side
length. Well, in that case, we can know
that we can’t be using the trigonometric ratios because if we’re gonna use the
Pythagorean theorem, we’d rather have two sides and try and find the further side or
something like that.
But here we’ve actually been given
an angle. And we’ve only got one side. Okay, so great! We know now to use the trig
ratios. To solve these kinds of problems, I
like to break it down to four clear steps.
So step one is labelling sides. Well, the first side we’re gonna
label is the hypotenuse. We know it’s the hypotenuse because
it’s opposite the right angle. And it’s also the longest side. Okay, great! Let’s label the next side. Well, the next side to label is the
opposite. And this is the opposite because
it’s opposite the angle that we’re given or that we’re trying to find. So we can see that that’s this one
here, which is 2.44 meters. And then finally, we label the
adjacent. Okay, great! Step one complete. All our sides are labelled.
Okay, now we can move on to step
two. And step two is choosing the
ratio. So we’re gonna choose which one of
our trigonometric ratios we’re actually going to use. To assist with this, what we’ll
actually use is a mnemonic, which is SOHCAHTOA. This helps us remember how we
actually find our sine, cosine, and tangent ratios.
Okay, so now let’s choose which one
we’re going to use. Okay, so what I’ve done is I’ve
actually looked at which side we have. So we’re given the opposite because
it’s 2.44 meters. And then the side we’re looking for
in this question is actually the adjacent because we want to find 𝑋, which is the
distance between the footballer and the goal.
Okay, fantastic! So now we’ve got these two. Let’s use SOHCAHTOA to help us
decide which ratio to use. Well, if we look back at SOHCAHTOA,
we can see that it’s the final one, the TOA part, that contains both the opposite
and the adjacent. So therefore, we know we can use
the tangent ratio. And then we use our mnemonic to
remind us that tan of any angle is equal to the opposite divided by the
adjacent. Okay, fantastic! Step two done. We’ve chosen our ratio.
Okay, so now we move on to step
three. And step three is where we actually
substitute in the values into our formula for tan 𝜃. So therefore, we get that tan 38
degrees is equal to 2.44 divided by 𝑋. So great! We can move on to the final step,
step four, which is we’re gonna rearrange and solve.
So first of all, we’re actually
gonna multiply both sides by 𝑋. So we’re gonna get 𝑋 tan 38 is
equal to 2.44. And then the next step is to divide
by tan 38. So we get 𝑋 is equal to 2.44 over
tan 38. So therefore, this gives us that 𝑋
is equal to 3.123057, et cetera.
So great! We now calculated 𝑋. We just need to do the final part
of the question because we need to round, as our question wants the answer to two
decimal places. So great! We can therefore say that the
horizontal distance 𝑋 between the footballer and the goal is 3.12 meters to two