Final answer:
The slope of a ramp in trigonometry is the tangent of the angle that the ramp makes with the horizontal, calculated as the rise over the run. It corresponds to the 'm' in the linear equation y = mx + b.
Step-by-step explanation:
The slope of a ramp using trigonometry can be defined as the tangent of the angle that the ramp makes with the horizontal. Using the properties of right triangles, we can relate the slope to the sides of the triangle formed by the ramp, the height it rises (opposite side), and the horizontal distance (adjacent side) it covers. The slope (m) is calculated as the rise divided by the run, which corresponds to the tangent of the angle (θ) in trigonometric terms. In the equation y = mx + b, m represents the slope, and b is the y-intercept.
For instance, if a ramp rises 3 meters over a horizontal distance of 4 meters, its slope would be 3/4, which is the tangent of the ramp's incline angle. You can use trigonometric functions such as sine (sin), cosine (cos), and tangent (tan) to express these relationships more formally when given an angle or the lengths of sides of the right triangle.
To find the slope at a specific point on a curve, you look for the slope of the tangent line at that point, just as it's shown in Figure 2.48 where Q is the point at t = 25 s. Trigonometry can also be used to determine the components of vectors, as shown when determining the magnitude of the weight components on an incline which forms a right angle with the three weight vectors.