A 23-foot ladder leans against a building such that the angle between the ground and the ladder is 80 degrees. How high does the ladder reach up the side of the building?
So here we have a ladder that’s leaned up against the building. The ladder itself is 23 feet long and the angle between the ground and the ladder is 80 degrees. And we want to know how high does the ladder reach up the side of the building? So we can call that 𝑥.
Now, this is a well-made house; the house should be completely perpendicular with the ground, creating a 90-degree angle. So to read through the triangle that we have, we have 𝑥 as a side length, 23 would be the hypotenuse, the longest side because it’s across from the 90-degree angle, and then we have an 80-degree angle. We can use the trigonometric ratios in order to find this missing side. So first let’s go ahead and label each side.
Across from our angle will be the opposite side. As we stated before, 23 is the hypotenuse because it’s the side that is directly across from the 90-degree angle, which means it’s the longest side. And then we actually don’t need the last side, which is the adjacent side. So we wanna use opposite and hypotenuse. So to use this little thing Soh-Cah-Toa, S stands for the sine- so using a little thing Soh-Cah-Toa, S is for sine, C is for cosine, and T is for tangent. So out of the three, which one uses opposite and hypotenuse, o and h?
So the sine of 𝜃, and 𝜃 is just a general thing that we use when talking about angles, so we can actually replace 𝜃 with 80 degrees because we know we’re using 80 degrees. And then we can replace the opposite and the hypotenuse with 𝑥 and 23. And now we can solve for 𝑥. To do so, let’s multiply both sides by 23. So in our calculators, we will type in 23 times the sine of 80 degrees and we get that 𝑥 is equal to 22.65. So this means that the ladder reaches 22.65 feet up the side of the building.