91Ó°ÊÓ

When you hear 'tangent of an angle', you might imagine it has something to do with touching or reaching out. In trigonometry, it's a bit like that, but specific to a right triangle. Imagine you're looking at a triangle with one 90-degree angle—this is your right triangle. Now, if one of the other angles, let's say Angle A, is the one you're interested in, the tangent of Angle A is simply the length of the side opposite to it divided by the length of the side right next to it, which is not the hypotenuse. In a formula, this is written as \(tan(A) = \frac{opposite}{adjacent}\).
Think of it as a way to measure how steep a line is. For the problem concerning the road climbing at an 8-degree angle, we consider this road as the opposite side of a right triangle, and the road's extension on the flat ground as the adjacent side. You can find the tangent using a scientific calculator, or if you're feeling old school, a trigonometry table.

In our daily lives, we might not realize it, but trigonometry shows up everywhere, from the way a ramp is inclined to how a roof is pitched. It’s all about relationships between the angles and sides of triangles, especially right triangles. Right triangles are the bread and butter of trigonometry, and they have three parts you should know about: the hypotenuse (the longest side), the opposite side (across from the angle of interest), and the adjacent side (next to the angle of interest).
With these, there are three primary trigonometric functions: sine, cosine, and tangent. Each function relates to a specific ratio of the sides of the triangle. Since we're focused on tangent and grading roads, remember that it deals with the opposite and adjacent sides. It's also worth noting that trigonometry in right triangles has a myriad of applications, from architecture to navigation.

Now, onto the real-world application of this math—calculating the grade percentage! Picture the road winding up a hill; that road is creating an angle with the flat ground. The grade of the road tells us how steep the road is. More technically, it's the vertical change (rise) compared to the horizontal ground (run), just like the tangent we talked about. So, to calculate the grade percentage, just take the tangent of the road's angle, which gives you the 'rise over run' ratio, and multiply that number by 100 to get a percentage.
For instance, if we calculate the tangent of that 8-degree angle and find out it's 0.14, express that as a percentage by multiplying by 100. You'll find that the road's grade is 14%. Remember, a steeper road means a higher grade percentage, just like a steeper triangle side means a bigger tangent ratio.