### 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 trig ratios to solve for the two missing sides,
remembering the acronym SOHCAHTOA. 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
~~hypotenuse~~
[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. The 𝑦 is the opposite side to the
40-degree angle. The 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 in
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, which means we’ll round down. 𝑦 is equal to 8.999. And the units 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 side, 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. And that means we’ll have 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.