Video Transcript
Find the value of 𝑥 in the given
figure.
Let’s have a look at the diagram
that we’ve been given. It consists of a triangle. And we can see that this is a right
triangle. We know this because of the right
angle marked here. We’ve also been given the size of
one other angle in this triangle — it’s 30 degrees — and the length of one side in
this triangle — it’s five units. The value 𝑥, which we’ve been
asked to calculate, is the length of a second side in this triangle.
Now, as we have a right triangle in
which we know the size of one other angle and the length of one side and want to
calculate the length of a second side, we can apply trigonometry to this
problem. Trigonometry tells us about the
relationships that exist between the lengths of sides in right triangles. We’ll begin by labeling the three
sides of this triangle. The side opposite the right angle,
which is always the longest side in a right triangle, is called the hypotenuse. The side directly opposite the
known angle — in this case, that’s the side opposite the angle of 30 degrees — is
called the opposite. And the side which is between the
known angle and the right angle is called the adjacent.
In this question, the side we know
and the side we want to calculate are the hypotenuse and opposite of this
triangle. So, we’re interested in the
relationship that exists between these two lengths. We can use the memory aid SOH CAH
TOA to help us decide which of the three trigonometric ratios sin, cos, or tan we
should be using in this question.
Here, S, C, and T stand for sin,
cos, and tan. And O, A, and H stand for opposite,
adjacent, and hypotenuse. As the two sides that we’re
interested in are the opposite and hypotenuse, we’re going to be using SOH. That’s the sine ratio. Let’s recall its definition. The sine ratio tells us that sin of
an angle 𝜃 is equal to the ratio of the opposite side divided by the
hypotenuse.
In our triangle, the angle 𝜃 is 30
degrees, the length of the opposite side is 𝑥, the value we want to find, and the
length of the hypotenuse is five units. So, substituting each of these
values, we have the equation sin of 30 degrees is equal to 𝑥 over five.
Now, here’s a key fact about angles
of 30 degrees which we need to know. Sin of an angle of 30 degrees is
equal to one-half. This means that, in a right-angled
triangle with an angle of 30 degrees marked, the ratio of the opposite side and the
hypotenuse is always one-half. Which in practical terms means that
the hypotenuse is always twice as long as the opposite.
You may not actually have learned
about sine, cosine, and tan formally yet, but instead you may have learned that in
any right-angled triangle with an angle of 30 degrees, the ratio between the
opposite and hypotenuse is one-half. So, by substituting this value of
one-half for sin of 30 degrees or just recalling that the ratio between these two
sides is one-half, we get the equation one-half is equal to 𝑥 over five.
To solve this equation for 𝑥, we
need to multiply both sides of the equation by five, leaving 𝑥 on its own on the
right-hand side and giving five over two on the left-hand side. We can give our answer as a
fraction if we wish, or we can convert it to a decimal. Five divided by two, or half of
five, is 2.5. So, the value of 𝑥 in this
triangle then, found by either recalling that the ratio between the opposite and
hypotenuse of a right-angled triangle with an angle of 30 degrees is one-half or by
formally solving the equation sin of 30 degrees equals 𝑥 over five, is 2.5.
