Right Triangles

Ramps, ladders and ropes often sit at an angle. To work with a slanted length you split it into a part along the ground and a part straight up, and the tool for that is the right triangle.

The sides are named from the angle θ: the side next to it is the adjacent side, the side across from it is the opposite side, and the longest side is the hypotenuse.

The sides are named from the angle θ.

Two rules connect them:

$\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}, \quad \cos\theta = \frac{\text{adjacent}}{\text{hypotenuse}}$

Multiply both sides by the hypotenuse and you get the one you will use most: opposite = hypotenuse × sin θ.

Example: a ramp L = $8\ \mathrm{m}$ long rises at $30^\circ$. Its height h is the side opposite the angle, and the ramp is the hypotenuse. The rule in letters first:

$h = L \sin 30^\circ$

Then the numbers: h = 8 × 0.5 = $4\ \mathrm{m}$. The values you will use most: sin $30^\circ$ = 0.5, cos $60^\circ$ = 0.5 and sin $90^\circ$ = 1; the calculator has the others.