Which ratio of sides is the tangent of a marked angle?
A trail climbs 3 m for every 4 m it runs. How steep is that, as an angle?
We know two sides of the right triangle. This time we want the marked angle itself.
Divide the rise by the run, and you have the angle's tangent: tan θ = opp / adj = 3 / 4 = 0.75. So θ is the angle whose tangent is 0.75.
A calculator finds that angle with the inverse tangent, written tan⁻¹. On most calculators, tan⁻¹ sits above the tan key and is reached with the 2nd or shift key.
θ = tan⁻¹(0.75) ≈ 37°. The trail climbs at about 37°. tan⁻¹ undoes tan: tan 37° ≈ 0.75, and tan⁻¹(0.75) ≈ 37°.
A ramp rises 2 m over a 5 m run. Calculate the angle the ramp makes with the ground, to the nearest degree.
Write down the values in the question:
opp = 2 m, adj = 5 m, θ = ?
Write down the equation:
θ = tan⁻¹(opp / adj)
Substitute in the values:
θ = tan⁻¹(2 / 5) = tan⁻¹(0.40)
θ ≈ 22°.
A mast rises 33 m, and a stripe painted on the ground reaches 16 m from its base. Calculate the angle between the ground and the line from the stripe's end to the mast's top, to the nearest degree.
Write down the values in the question:
opp = 33 m, adj = 16 m, θ = ?
Write down the equation:
θ = tan⁻¹(opp / adj)
Substitute in the values:
θ = tan⁻¹(33 / 16) = tan⁻¹(2.06)
θ ≈ 64°: the ratio 2.06 is past 1, and the angle is steep.
Now make the move quick. Six items follow: divide opposite by adjacent, then press tan⁻¹.
A ramp rises 8 m over an 8 m run. Enter its climb angle in degrees, to the nearest degree. Answers within 1° are accepted.
A driveway rises 6 m over a 10 m run. Enter its climb angle in degrees, to the nearest degree. Answers within 1° are accepted.
A ladder's top sits 12 m up a wall, its foot 5 m out from the wall. Enter the angle between the ladder and the ground, to the nearest degree. Answers within 1° are accepted.
A wheelchair ramp rises 9 m over a 25 m run. Enter its climb angle in degrees, to the nearest degree. Answers within 1° are accepted.
A roofline rises 15 m over a 10 m run. Enter its angle to the horizontal, to the nearest degree. Answers within 1° are accepted.
A drainage pipe drops 7 m over a 28 m run. Enter its angle below the horizontal, to the nearest degree. Answers within 1° are accepted.
A conveyor rises 11 m over a 16 m run. Enter its climb angle in degrees, to the nearest degree. Answers within 1° are accepted.
A ski jump's takeoff rises 27 m over a 12 m run. Enter its angle to the horizontal, to the nearest degree. Answers within 1° are accepted.
In a right triangle, the marked angle's opposite side is 13 m and its adjacent side is 30 m. Enter the marked angle in degrees, to the nearest degree. Answers within 1° are accepted.