Video Transcript
Find the values of 𝑥 and 𝑦,
giving the answer to three decimal places.
We notice that this is a right
triangle. We’re given an angle and a side
length, which means we can use trigonometric ratios to solve for the two missing
sides. Remembering the acronym SOH CAH
TOA, sin of 𝜃 equals the opposite over the hypotenuse, cos of 𝜃 equals the
adjacent over the hypotenuse, and tan of 𝜃 equals the opposite over the
adjacent. The key here is for us to label
this triangle correctly. And to do that, we’ll use the given
angle as our starting point.
We label the side lengths relative
to our given angle. 𝑦 is the side opposite to the
40-degree angle. 𝑥 is the adjacent side to the
40-degree angle. And the hypotenuse is always the
side opposite the right angle.
First, let’s try to solve for
𝑦. If we’re solving for 𝑦 and we know
the hypotenuse, we’ll use the sine ratio because the sin of 𝜃 is the opposite over
the hypotenuse. And that means we can say that the
sin of 40 degrees is equal to 𝑦 over 14. Since our goal is to solve for 𝑦,
we’ll multiply both sides by 14. And then we’ll see that 14 times
sin of 40 degrees equals 𝑦. When we plug that into our
calculator, we get 8.99902 continuing. If you don’t get this answer on
your calculator, then you should check and make sure that you’re operating in
degrees and not in radians.
We want our answer to three decimal
places. So we look to the fourth decimal
place, where there is a zero. That means we’ll round down. 𝑦 is equal to 8.999. And the units that we’re measuring
are centimeters. So we say that 𝑦 equals 8.999
centimeters.
Next, we need to solve for 𝑥. And we can solve for 𝑥 with two
different ratios. We could use the adjacent side and
the hypotenuse, which would be the cosine ratio. Or we could take what we found for
𝑦 and use that as the opposite side. And that would mean we would use
the tangent ratio because we would have the opposite and adjacent sides. In this case, let’s use the
hypotenuse as it will save us a little bit of writing.
We’re dealing with the cosine
ratio. We have cos of 40 degrees is equal
to 𝑥 over 14. We’ll multiply both sides by
14. 14 times cos of 40 degrees will
equal 𝑥. So 𝑥 will be equal to 10.72462
continuing. Rounded to the third decimal place
means we need to round up to 10.725. Again, the units here will be
measured in centimeters. And so we can say that 𝑥 is equal
to 10.725 centimeters and 𝑦 is equal to 8.999 centimeters, each to three decimal
places.