[eli5] why is gradient = tangent theta?

Question

[eli5] why is gradient = tangent theta?

In: 0

Answer 1

Because the change in the y direction is given by r sin(t) and the change in the x direction is given by r cos(t). t is the angle of the right triangle whose hypotenuse is the line segment connecting the two points and whose legs are parallel to the x and y axis, respectively, and r is the length of that hypotenuse.

gradient = change in y / change in x

= (r sin(t)) / (r cos(t))

= sin(t) / cos(t)

= tan(t).

Answer 2

tan(θ) is defined as sin(θ)/cos(θ). Which could also be rewritten as “rise”/”run”, since sin(θ) represents the vertical component of the angle, and cos(θ) represents the horizontal component. And rise/run is the description of a slope or gradient.