A zip line: cable 13 m, vertical drop 5 m, ground run 12 m, the angle marked at the bottom. Which name fits the 5 m drop?
The skateboard ramp again: rise 3 m, run 4 m, slope 5 m, with the bottom angle marked. That angle is about 37°.
The ramp's right triangle hides three numbers that depend only on the marked angle.
On the ramp, divide the opposite side by the hypotenuse: 3 / 5 = 0.60. Now scale the ramp up: rise 6 m, run 8 m, slope 10 m. The marked angle is still about 37°. The division comes out at 6 / 10 = 0.60 again.
The ratio opposite over hypotenuse is fixed by the marked angle alone. It is called the sine of the angle, written sin θ.
The ratio adjacent over hypotenuse is called the cosine, written cos θ. On the ramp, that ratio is 4 / 5 = 0.80. The ratio opposite over adjacent is called the tangent, written tan θ. On the ramp, that ratio is 3 / 4 = 0.75.
Which ratio of sides is the sine of a marked angle?
Each ratio is one length divided by another, so the units cancel: sin, cos and tan are plain numbers with no unit.
A wire runs 13 m from a pole's top to the ground, meeting the ground 12 m from the pole's base; the pole is 5 m tall. The angle is marked where the wire meets the ground. Determine sin θ, cos θ and tan θ.
Name the sides:
hypotenuse 13 m (the wire), opposite 5 m (the pole), adjacent 12 m (the ground stretch)
Write down the equations and divide:
sin θ = opp / hyp = 5 / 13 ≈ 0.38
cos θ = adj / hyp = 12 / 13 ≈ 0.92
tan θ = opp / adj = 5 / 12 ≈ 0.42
sin θ ≈ 0.38, cos θ ≈ 0.92, tan θ ≈ 0.42.
Mark the ramp's top angle instead, about 53°. Its opposite side is now the 4 m run, and its adjacent side is the 3 m rise. So tan 53° = 4 / 3 ≈ 1.33. A tangent can pass 1. A sine or cosine never passes 1, because the hypotenuse is always the longest side.
Now make the reads quick. Four items follow: build each ratio straight from the stated sides. A fifth then asks which ratio is which.
A ramp: rise 21 m (opposite the marked angle), slope 29 m (hypotenuse). Enter sin θ to 2 decimal places. Answers within 0.01 are accepted.
A ladder triangle: ground stretch 24 m (adjacent to the marked angle), ladder 25 m (hypotenuse). Enter cos θ to 2 decimal places. Answers within 0.01 are accepted.
A kite triangle: height 8 m (opposite the marked angle), ground stretch 15 m (adjacent). Enter tan θ to 2 decimal places. Answers within 0.01 are accepted.
A cliff rises 35 m (opposite the marked angle) over a 12 m base (adjacent). Enter tan θ to 2 decimal places. Answers within 0.01 are accepted.
Which ratio of sides is the tangent of a marked angle?
A 53 m wire runs from the top of a 45 m tower to an anchor 28 m from its base. The angle is marked at the anchor. Enter sin θ to 2 decimal places. Answers within 0.01 are accepted.
A ramp rises 1.2 m over a 3.5 m run; the angle is marked at the bottom. Enter tan θ to 2 decimal places. Answers within 0.01 are accepted.
A driveway climbs 9 m over a 40 m run; its slope is 41 m. The angle is marked at the bottom. Which statement is correct?