The base of a truck is two feet off the ground and a 20-foot ramp is used to load the truck. Find, to two decimal places, the angle the ramp makes with the horizontal ground.
So, what I’ve done, to help us answer this question, is drawn a sketch. So, we can see that we’ve got our ramp, which is 20 foot, and the base of our lorry is two foot above the ground. And what we’re trying to do is find the angle 𝜃 because that is the angle that the ramp makes with the horizontal ground.
So, I’ve enlargened our sketch of the triangle, so we can see what we need to do. So, we’ve got a right-angle triangle. So therefore, we think, well, what kind of maths can we use? We could use the Pythagorean theorem because it’s a right-angle triangle. However, we’re looking to find an angle not the third side. So, the Pythagorean theorem wouldn’t be the correct thing to use. So, what we can, therefore, use is the trig ratios, or trigonometric ratios.
And what we use to help us remember the trigonometric ratios, or the trig ratios, is this memory aid SOH CAH TOA. There are others that you can use. This is just the one that I seem to remember the best. And what this does is it helps us remember that the sine of an angle is equal to the opposite divided by the hypotenuse, so O and H. The cosine of an angle is equal to the adjacent divided by the hypotenuse. And the tangent of an angle is equal to the opposite divided by the adjacent.
So, when I’m using the trig ratios, what I like to do is use a four-step process. Step one, label the sides. So, first of all, we have the hypotenuse. And this is the longest side opposite the right angle. Then, we have the opposite which is the side opposite the angle we’re given, or looking for, in this case 𝜃. And then, we have the adjacent which is the side next to the angle and it’s between the angle and the right angle.
So, that’s step one complete. We now move on to step two. We need to choose the correct ratio. So, what I’ve done is circled in our the case the two sides that we’ve been given. So, we’ve got the hypotenuse, cause that’s 20 foot, and the opposite, cause that’s two foot. In a question where you’re looking to find one of the sides, then you’d choose the side that you’re looking for and the side you’ve been given.
So, then we can use our memory aid, so SOH CAH TOA. So, we can see that the first part SOH has got O and H. So therefore, this is the ratio we’re going to use. So, that tells us that sin 𝜃 is equal to the opposite divided by the hypotenuse. So, great, that’s step two complete. We can now move on to step three. And step three is substitute because we’re gonna substitute our values into sin 𝜃 is equal to the opposite divided by the hypotenuse.
So, when we do that, we get sin 𝜃 is equal to two over 20. And that’s cause two is our opposite and 20 was our hypotenuse. This is why we label early on in the question, so that we make sure we get the correct sides when we are substituting them in. So, now that step three is complete, we can move on step four. And that’s the final step. We’re gonna solve to find 𝜃.
So, first of all, we can actually simplify two over 20 by dividing numerator and denominator by two. And when we do that, we get sin 𝜃 is equal to 0.1. And that’s because it’ll be one-tenth, and one-tenth is 0.1. So, we have sin 𝜃 is equal to 0.1, but this isn’t what we want because we want to know what 𝜃 is. So, to find 𝜃 what we do is we take the inverse sine of both sides of the equation. So, sin to the power of negative one.
So, you’ll see this on your calculator, and it’s often called shift sine. And that’s because you press the shift button then the sine button and then this will come up. So therefore, to find out the value of 𝜃, what you type into the calculator is the inverse sine and then of 0.1. So, this gives us 𝜃 is equal to 5.739 et cetera degrees. So, we’re just gonna round this to a sensible degree of accuracy.
So, I’m going to round it to two decimal places. So, we’re gonna get 5.74. And that’s because if you look at 5.739 et cetera, the second decimal place is three. So then, we look to the right of this. And we’ve got the deciding digit, or deciding number, and that’s nine. And because this is five or above, we round the three to a four.
So therefore, we can say that if the base of a truck is two foot off the ground and a 20-foot ramp is used to load the truck, the angle the ramp makes with the horizontal ground is equal to 5.74 degrees. And that’s to two decimal places. And if we double-check with the question, that’s the level of accuracy that it asked for.