Video Transcript
A car is going down a ramp which is 10 meters high and 71 meters long. Find the angle between the ramp and the horizontal, giving the answer to the nearest second.
In order to answer this question, we will begin by modeling the ramp as a right triangle as shown. We are told that the ramp is 10 meters high. It is 71 meters long. We are asked to calculate the angle between the ramp and the horizontal, which we will call 𝜃.
We can now answer this question using our knowledge of the trigonometric ratios in right triangles. These state that the sin of angle 𝜃 is equal to the opposite over the hypotenuse. The cos of 𝜃 is the adjacent over the hypotenuse. And the tan of 𝜃 is equal to the opposite over the adjacent. One way of remembering these ratios is using the acronym SOH CAH TOA.
We know that the longest side of a right triangle, which is opposite the right angle, is known as the hypotenuse. The opposite side is the one that is opposite the angle we are dealing with. In this question, it is the height of the ramp. The adjacent side joins the angle we are dealing with and the right angle. In this question, we know the length of the opposite and hypotenuse. So we will use the sine ratio.
Substituting in our values, we have the sin of 𝜃 is equal to 10 over 71. Taking the inverse sine of both sides of this equation, we have 𝜃 is equal to the inverse sin of 10 over 71. Ensuring that our calculator is in degree mode, we can type this in, giving us 𝜃 is equal to 8.09674 and so on degrees.
We are asked to give our answer to the nearest second. And one way to do this is to simply press the degrees, minutes, seconds key on our calculator. This gives us eight degrees, five minutes, and 48.3 seconds, which rounded to the nearest second is 48 seconds. Alternatively, we can multiply the decimal part of our answer, 0.09674 and so on, by 60. This is because there are 60 minutes in a degree and gives us 5.8049 and so on minutes. We can then multiply the decimal part of this answer by 60 to give us the number of seconds. The angle between the ramp and the horizontal is eight degrees, five minutes, and 48 seconds.
