Which ratio of sides is the sine of a marked angle?
A zip line runs 50 m from a platform to the ground, descending at 37° below the horizontal. How far does a rider drop?
We know one side, the 50 m cable, and the marked angle. We want another side. The ratios from the last lesson can produce it.
sin 37° is a fixed number: opposite over hypotenuse in every right triangle whose marked angle is 37°. A calculator stores that number. Set the calculator to degrees mode, type sin 37, and the calculator shows about 0.60.
The routine starts by naming the sides. The drop is the marked angle's opposite side. The cable is the hypotenuse. Pick the ratio built from the side you know and the side you want: opposite with hypotenuse means sine. Write the ratio as an equation: sin θ = opp / hyp. Make the unknown the subject: opp = hyp × sin θ.
A zip line cable is 50 m long and descends at 37°. Calculate the rider's vertical drop. Take sin 37° = 0.60.
Write down the values in the question:
hyp = 50 m, θ = 37°, opp = ?
Write down the equation:
sin θ = opp / hyp
Make opp the subject:
opp = hyp × sin θ
Substitute in the values:
opp = 50 × sin 37° = 50 × 0.60 = 30
opp = 30 m: the rider drops about 30 m.
A playground slide is 16 m long and rises at 30°. Enter its vertical height in m.
If the known side and the wanted side are opposite and adjacent, use the tangent. If the pair is adjacent and hypotenuse, use the cosine.
A 30 m wire meets the ground at 60°. Enter the distance from the anchor to the pole's base in m.
A ramp climbs at 37° over a 28 m horizontal run. Take tan 37° = 0.75. Enter the rise in m.
Sometimes the unknown sits on the bottom of the ratio. Then the rearrangement divides: hyp = opp / sin θ.
A kite flies 12 m above the ground, and its string makes a 30° angle with the ground. Calculate the string's length.
Write down the values in the question:
opp = 12 m, θ = 30°, hyp = ?
Write down the equation:
sin θ = opp / hyp
Make hyp the subject:
hyp = opp / sin θ
Substitute in the values:
hyp = 12 / sin 30° = 12 / 0.5 = 24
hyp = 24 m: the string is 24 m long.
Now make the move quick. Three more follow: name the sides, pick the ratio, multiply.
A 45 m zip line descends at 53°. Take sin 53° = 0.80. Enter the vertical drop in m.
A 55 m rope anchors a balloon, meeting the ground at 37°. Take cos 37° = 0.80. Enter the ground distance from anchor to the point below the balloon, in m.
A 90 m trail climbs steadily at 37°. Take sin 37° = 0.60. Enter the height gained in m.
A 70 m zip line descends at 30°. Enter the rider's vertical drop in m.
Jonas's kite flies 26 m above the ground, and its string makes a 30° angle with the ground. Enter the string's length in m.
An 85 m cable runs from a tower's top to a ground anchor, meeting the ground at 37°. Take cos 37° = 0.80. Enter the distance from the anchor to the tower's base in m.