Video Transcript
Find 𝑥 to two decimal places.
In this right triangle, we know the
measure of one of the other angles and the length of one side. We want to calculate the length of
another side of this triangle. We can do this using
trigonometry. We’ll begin by labeling the three
sides of this triangle in relation to the angle of 20 degrees. The side directly opposite the
right angle is the hypotenuse of the triangle. The side opposite the angle of 20
degrees is the opposite. And the side between the right
angle and the angle of 20 degrees is the adjacent.
Next, we recall the acronym
SOHCAHTOA to help us decide whether we need the sine, cosine, or tangent ratio in
this question. The side we know is the
opposite. And the side we want to calculate
is the hypotenuse. So we’re going to be using the sine
ratio. For a given angle 𝜃 in a right
triangle, the sine ratio, sin of 𝜃, is equal to the length of the opposite side
divided by the length of the hypotenuse. We can substitute the values for
this triangle into this definition. 𝜃 is equal to 20 degrees, the
opposite is 12 units, and the hypotenuse is this unknown 𝑥. So we have the equation sin of 20
degrees is equal to 12 over 𝑥.
Now, we must be careful here. A really common mistake is to think
that the unknown, in this case 𝑥, must always be in the numerator of the fraction,
and so to write down instead sin of 20 degrees is equal to 𝑥 over 12. But of course, if we did that, we
would be dividing the length of the hypotenuse by the length of the opposite, not
the length of the opposite by the length of the hypotenuse. This is a really common mistake,
though. So we just need to take our time
when substituting the values or expressions for each side of the triangle into the
definition of our trigonometric ratios. We now need to solve this equation
where 𝑥 appears in the denominator of the fraction, and this will require two
steps.
First, we multiply both sides of
the equation by our unknown 𝑥. On the left-hand side, we now have
𝑥 sin 20 degrees and on the right-hand side, 12 over 𝑥 multiplied by 𝑥 simplifies
to 12. Next, we need to divide both sides
of the equation by sin of 20 degrees. Remember, this is just a number, so
it’s absolutely fine to do this. This gives 𝑥 is equal to 12 over
sin of 20 degrees. Finally, we evaluate on our
calculators, giving 35.085. Remember, we must make sure that
our calculators are in degree mode in order to give the correct answer. The question specifies that we
should give our answer to two decimal places. So we round to 35.09. So by applying the sine ratio in
this right triangle, we found that the value of 𝑥 to two decimal places is
35.09.
