Given the following figure, find the lengths of 𝐴𝐵 and 𝐵𝐶 and the measure of angle 𝐴𝐵𝐶 in degrees. Give your answers to two decimal places.
Our question is asking us for three different pieces of information: the length of 𝐴𝐵 here, the length of 𝐵𝐶 here, and the measure of angle 𝐴𝐵𝐶 here. Let’s start by finding the length of side 𝐴𝐵. To do that, we should ask what do we know? We know an angle measure of 58 degrees and we know an adjacent side length to that angle. Side length six, 𝐴𝐶, is adjacent to angle 𝐶.
Our next question would be what do we want to know? what is missing? The missing side is the side opposite angle 𝐶. What I want you to see here is we have an angle, an adjacent side length, and an opposite side length. Which trig function would fit here? What would help us solve this?
The tangent ratio deals with an angle, an opposite side length, and an adjacent side length. And this is all the information that we have. We have tangent of 58 degrees equals our missing side length over six. We can treat our missing side length like a variable and solve for it. To get side length 𝐴𝐵 by itself, I’ll multiply both sides of our equation by six over one. On the right side of our equation, divide by six. Multiplying by six cancels each other out, leaving you with side length 𝐴𝐵.
On the other side, we have six times tangent of 58 degrees. If we plug that into the calculator, we get 9.60200. Our question is asking us to give our answers to two decimal places. So we look at the thousands place to see how we’ll round, and we find out that side length 𝐴𝐵 equals 9.60 units. So we can plug that information into our picture.
From there, we can move on to finding the length of side 𝐵𝐶. Because we’re working with a right triangle and we already know two of the sides, we can use the Pythagorean theorem to find the length of our hypotenuse, our missing third side. Here, we plug in the two shorter sides, 𝐴 and 𝐵. We’ll have six squared plus 9.6 squared equals 𝐶 squared. Six squared equals 36; 9.6 squared equals 92.16. When we add those two together, we get 128.16 equals 𝐶 squared.
From there, we take the square root of both sides of our equation, and our calculator tells us that 𝐶 would be equal to 11.3207. But again, we want to round to two decimal places. So we check our hundreds position and then we round to 11.32. We can go ahead and label our graph. Side length 𝐵𝐶 is 11.32 in length.
We have one last missing piece of information; we want to know what is the measure of angle 𝐴𝐵𝐶, this angle. We know that when we add up all angles in a triangle, they’ll equal 180 degrees. And since we have two out of those three, we can plug that in here and solve for angle 𝐵. Angle 𝐴 is our right angle, which means it’s 90 degrees, angle 𝐵 is our missing angle, and angle 𝐶 is 58 degrees.
If we add 90 and 58, we get 148 plus angle 𝐵 will equal 180 degrees. Subtract 148 degrees from both sides of the equation. 148 minus 148 cancels out. And then we find that angle 𝐵 equals 32 degrees. But remember that all of our answers need to be given to two decimal places. So for angle 𝐵, we’ll say 32 and zero hundreds, 32.00.