Video Transcript
Find the values of 𝑥 and 𝑦,
giving the answer to three decimal places.
We have a right triangle. We’re given an angle and a side
length and asked to find the two missing sides. To do this, we’ll need our
trigonometric ratios. And to remember those, we’ll use
SOH CAH TOA. The sin of 𝜃 equals the opposite
over the hypotenuse. The cos of 𝜃 equals the adjacent
over the hypotenuse. And the tan of 𝜃 equals the
opposite over the adjacent. The key to solving these problems
consistently is to correctly label the triangle. And we label them relative to the
angle that we’re given. This is our starting point. The opposite side length is the
side length directly opposite this angle. The adjacent side is between this
angle and the right angle. And the hypotenuse is always
opposite the right angle.
Once a triangle is labeled, we’re
ready to identify which of the ratios we need. If we start by finding side length
𝑦, the hypotenuse, and we already know the opposite side, 28 centimeters, we need
to use the sine ratio, as the sine ratio involves the opposite side length and the
hypotenuse. The ratio would look like this. sin of 47 degrees is equal to 28
over 𝑦. When our variable is in the
denominator, it will take two steps to find the value.
The first thing we would do is
multiply both sides of the equation by 𝑦. When we do that, we get 𝑦 times
sin of 47 degrees equals 28. If the goal is to isolate 𝑦, then
at this point, we need to divide both sides of the equation by sin of 47
degrees. And then on the left we’ll just
have 𝑦, and on the right we’ll have 28 over sin of 47 degrees.
When we plug that into the
calculator, we get 38.28516 continuing. We need to round it to three
decimal places. This value rounds down to
38.285. The sides are being measured in
centimeters, so the units here would be centimeters. And that means we found one of the
missing sides.
To find the side length 𝑥, we’ll
have two choices. We could use the hypotenuse we just
found, 38.285. If we did that, we’d be dealing
with the adjacent side and the hypotenuse, which would be the cosine
relationship. Or we could use the 28-centimeter
side. In that case, we would be using the
opposite side and the adjacent side and would need the tangent ratio.
In this case, let’s practice having
the 𝑥-variable in the denominator. tan of 47 degrees equals 28 over
𝑥. To solve for 𝑥, we first multiply
both sides of the equation by 𝑥. Then, we can say that 𝑥 times tan
of 47 degrees equals 28. To isolate 𝑥, we divide both sides
of the equation by tan of 47 degrees. And so we say that 𝑥 equals 28
over the tan of 47 degrees, which gives us 26.11042 continuing. We round to the third decimal
place, and we get that 𝑥 is equal to 26.110. This is measured in
centimeters. And so we found the two missing
side lengths. To three decimal places, 𝑥 is
equal to 26.110 centimeters and 𝑦 is equal to 38.285 centimeters.
