A palm tree snaps due to bad winds. The vertical trunk is five meters tall, and the inclined part is six meters. Find the measure of the angle between the inclined part and the ground, giving the answer to the nearest second.
In order to answer this question, we will begin by sketching a right triangle to model the situation. We are told that the vertical trunk is five meters tall and that the inclined part of the tree is six meters. We have been asked to find the measure of the angle between the inclined part and the ground, which we have labeled 𝜃.
We will answer this question using our knowledge of the trigonometric ratios in right triangles. We know that sin 𝜃 is equal to the opposite over the hypotenuse. cos 𝜃 is equal to the adjacent over the hypotenuse. And tan 𝜃 is equal to the opposite over the adjacent. One way of recalling these is using the acronym SOH CAH TOA.
Going back to our diagram, we know that the longest side of our triangle, which is opposite the right angle, is known as the hypotenuse. The side that is opposite the angle we are working with is known as the opposite. In this question, the vertical trunk is the opposite side and the inclined part of the tree is the hypotenuse.
Since we know the length of both the opposite and hypotenuse, we can use the sine ratio. Substituting in our values, we have sin 𝜃 is equal to five over six or five-sixths. We can then take the inverse sine of both sides. 𝜃 is equal to the inverse sin of five over six. Ensuring that our calculator is in degree mode, this gives us an answer of 56.44269 and so on degrees.
We are asked to give our answer to the nearest second. And one way to do this is to use the degrees, minutes, and seconds button on our calculator. Pressing this key gives us 𝜃 is equal to 56 degrees, 26 minutes, and 33.68 seconds, which, rounding to the nearest second, is 34 seconds.
Alternatively, we can multiply the decimal part of our answer by 60 as there are 60 minutes in one degree. This gives us 26.5614 and so on minutes. We can then multiply the decimal part of this answer by 60, giving us 33.68 and so on seconds. Rounding to the nearest second, the measure of the angle between the inclined part of the tree and the ground is 56 degrees, 26 minutes, and 34 seconds.