Calculate the length 𝑥.
Looking at the diagram, we can see that we have a right-angled triangle in which we
want to calculate the length of one of the sides. When we want to calculate missing lengths in right-angled triangles, there are two
approaches that we can take. We can either use Pythagoras’s theorem or we can apply trigonometry, depending on the
other information that we’ve been given.
In order to apply Pythagoras’s theorem, we need to know two side lengths in a
right-angled triangle. But looking at the diagram, we can see that we only know one other side length, which
means we can’t apply Pythagoras’s theorem to this problem.
To apply right angle trigonometry to a problem like this, we need to know one other
side length and one angle other than the right angle, which we can see we do know
here. So we’ll answer this problem using trigonometry.
For me, the first step in any problem involving trigonometry is to label the three
sides of the triangle. So first, we have the hypotenuse, which is opposite the right angle. It’s the longest side of the triangle. Next, we have the opposite, which is the side opposite the angle we’ve been
given. In this case, that’s the side 𝑥, which is opposite the angle of 34 degrees. The final side, the adjacent, is between the angle we know and the right angle.
Next, we can use the memory aid SOHCAHTOA to help us decide which of the three
trigonometric ratios we need to use. Here, S, C, and T stand for sin, cos, and tan and O, A, and H stand for opposite,
adjacent, and hypotenuse, as in our labelling of the triangle. The side we want to calculate is the opposite, and the side we know is the
hypotenuse. And O and H appear together in the SOH part of SOHCAHTOA. So it’s the sine ratio that we need to use in this question. Let’s recall its definition.
The sine ratio tells us that sin of an angle 𝜃 is equal to the length of the
opposite side divided by the length of the hypotenuse. In this question, the angle 𝜃 is 34 degrees, the opposite is 𝑥, and the hypotenuse
is seven. So we substitute each of these values into the equation, giving sin of 34 degrees
equals 𝑥 over seven.
To solve this equation for 𝑥, we just need to multiply both sides by seven, as this
will eliminate the denominator on the right-hand side, leaving 𝑥 on its own. This gives seven multiplied by sin of 34 degrees is equal to 𝑥.
At this point, we can use our calculator to evaluate seven multiplied by sin of 34
degrees, making sure that our calculator is in degree mode. And it gives 3.91435.
We haven’t been asked to round our answer in a particular way. So our default level of accuracy should be to round to three significant figures. Here the fourth significant figure is a four, which tells us that we’re rounding
down. And so the one, the third significant figure, will remain a one. We also need to include units with our answer, which we can see from the diagram are
metres. So we found that the length 𝑥 to three significant figures is 3.91 meters.