Work out side length 𝑥.
So the first thing in this question that we need to do is work out what we’re going to use to solve it. So we ask ourself a couple of questions. First of all, has it got a right angle? Well, the answer is yes. So then we say has it got two sides that we know and we need to know the other side? Well, the answer to that is no, because we’ve got one side that we know and we need to find another side. So then we say has it got an angle? And the answer is yes. So now we know what we’re going to use is our trig ratios.
So if we’re using trigonometry to solve a problem, in this case find the length, I like to use set steps to do this. The first step is to label the triangle. So I start off by labeling the hypotenuse. And I know this is the hypotenuse because it’s the longest side and it’s opposite the right angle. Next, I’m going to label the opposite. And this is the side that’s opposite the angle that we’re either looking for or we have. And then, finally, we have the adjacent, which is the side next to our angle.
So now we’re gonna move on to step two. In step two, we need to choose our ratio. And a memory aid that we use to help us choose our ratio is something called SOHCAHTOA. Well, we’ve got SOH, which means that the sine of an angle is equal to the opposite divided by the hypotenuse; CAH, where the cosine of an angle is equal to the adjacent divided by the hypotenuse; and TOA, which means that the tangent of an angle is equal to the opposite divided by the adjacent.
So in our triangle, we can see that we’ve got the opposite, because that’s 11 centimeters, and we want to find the hypotenuse, which is 𝑥 centimeters. And because we have O and we’re trying to find H, we’re going to use the sine ratio, because SOH has O and H in it. So we can say that sine of the angle or sin 𝜃 is equal to the opposite divided by the hypotenuse.
So then we move on to step three. And step three is to substitute in the values that we have. So we have sin 42 is equal to 11 over 𝑥. And then, finally, we move on to step four. And step four is solve. So now we’re going to rearrange and solve our equation.
So the first step to solve is by multiplying each side of the equation by 𝑥. We do that because we’re gonna remove 𝑥 from the denominator of the fraction on the right-hand side. And when we’ve done that, we get 𝑥 sin 44 equals 11. And then, next, what we’re gonna do is divide each side of the equation by sin 42. We want to do that so that we’ve got 𝑥 on its own. So then we get that 𝑥 is equal to 11 divided by sin 42.
So we put this into our calculator. And we can say that, therefore, 𝑥 is gonna be equal to 16.439, et cetera. Therefore, we can actually now round it to a sensible degree of accuracy. So we’ve rounded it to one decimal place. So we can say that the side length 𝑥 is 16.4 centimeters.