𝐴𝐵𝐶 is a right triangle at 𝐵, where 𝐴𝐵 equals 26 centimeters and the measure of angle 𝐴 is 64 degrees. Find the length of 𝐴𝐶, giving the answer to two decimal places.
So we are told that we have a right triangle at 𝐵. And it’s triangle 𝐴𝐵𝐶, where 𝐴𝐵 is 26 centimeters and the measure of angle 𝐴 is 64 degrees. Find the length of 𝐴𝐶. So we can call it 𝑥.
Since we’re working with a right triangle, we can use the trigonometric identities SOHCAHTOA. Sine is equal to the opposite side divided by the hypotenuse side. Cosine is equal to the adjacent side divided by the hypotenuse side. And tangent is equal to the opposite side divided by the adjacent side.
So let’s go ahead and look at our right triangle and label the opposite side, the adjacent side, and the hypotenuse side. So all of these sides are referencing to a specific angle. And the only angle that we are given is angle 𝐴.
So the side exactly opposite of that angle would be the opposite side. And next, the hypotenuse is always directly across from the 90-degree angle. It’s the longest side. And then lastly, the adjacent side is next to the angle. But it’s not the hypotenuse.
So now that we have that labelled, the two sides that we’re interested in are 𝐴𝐵 and 𝐴𝐶, because 𝐴𝐵 we are given, and it’s 26 centimeters, and 𝐴𝐶 we’re trying to find. So that is the adjacent side and the hypotenuse side. So which identity uses adjacent and hypotenuse? It’s cosine.
So we take the cosine of the angle. So the cos of 64 degrees is equal to the adjacent side divided by the hypotenuse side. And the adjacent side we can replace with 26 centimeters. And the hypotenuse side we call 𝑥. So the cos of 64 degrees we can evaluate. We simply plug it into our calculator. And the cos of 64 degrees is equal to 0.43837114678.
So now we need to solve for 𝑥. So let’s go ahead and multiply both sides of the equation by 𝑥. They cancel on the right-hand side. So now to solve for 𝑥, we need to divide both sides of the equation by the decimal. It cancels on the left-hand side. And then on the right-hand side, dividing 26 and 0.43837114678 gives us that 𝑥 is equal to 59.3104728503 centimeters.
And it’s important not to round until the very end. This avoids rounding errors. So we’re supposed to round this decimal two decimal places. So we will either keep the one a one or round it up to a two. So we look at the number to the right. Since zero is less than five, we will keep the one a one. Therefore, 𝐴𝐶 is equal to 59.31 centimeters.