Find 𝑥 to two decimal places.
What are some things we know about this problem, simply by looking at the picture? We know
that we’re dealing with a triangle. More specifically, this is a right triangle. We know that it’s a
right triangle because of this symbol on our picture. We know that 𝑥 is the hypotenuse, the longest
side. We know that because the 𝑥 is the side that’s opposite the right angle. We’re given an angle
measure of 20 degrees. We’re also given the length of the side that is opposite the 20-degree
From here, we’re going to need some information that we don’t have simply by looking at the
picture. This is where trigonometry functions come in. The sine of some angle equals the opposite
side length over the hypotenuse side length. The cosine of an angle equals the adjacent side length
over the hypotenuse side length. And the tangent would be the opposite side length over the
adjacent side length. We’re given an angle measure. We’re missing the hypotenuse; we wanna
know what that value is. And we’re working with the side length that’s opposite the angle that
we’ve been given.
Since we have an opposite side length and the hypotenuse, we should use the sine function. If
we plug in our information into a sine function, it looks like this: sine of 20 degrees equals 12
over 𝑥. But remember, we’re solving for 𝑥, so we need to get 𝑥 by itself. To get 𝑥 by itself, we’ll first
need to get it out of the denominator here. We can multiply both sides of our equation by 𝑥
over one. On the right-hand side, we’re left with 12 because the 𝑥s cancel out. On the left-hand side,
we’ll have 𝑥 times sine of 20 degrees. 𝑥 is still not by itself, so we have one more step. We can divide
both sides of our equation by sine of 20 degrees. On the left side, sine of 20 degrees divided by sine
of 20 degrees equals one; they cancel out. 𝑥 is the only thing remaining.
What we have now is something that says the side length 𝑥 will be equal to 12 divided by the sine
of 20 degrees. If you plug that into your calculator, you should get something like this: 35.0856528.
But our question is asking us to find the length of 𝑥 to two decimal places. We can look at the third
decimal place to see if our eight will round up or stay the same. Because of the five, the eight will
round up leaving us with 𝑥 being approximately 35.09. We weren’t given what units, so we can say