Video: Using Right-Angled Triangle Trigonometry to Solve Word Problems Involving Angles of Elevation

A footballer kicks a ball into a goal post. The angle of elevation between the trajectory of the ball and the pitch is 38°. The ball hits the top of the goal post at a height of 2.44 m. Find the horizontal distance 𝑋 between the footballer and the goal, giving the answer to two decimal places.

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Video Transcript

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 trigonometric ratios.

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 decimal places.

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