Heights, Distances, & Slopes with Right-Triangle Trig

Use tangent to relate height and shadow: tan(angle) = opposite / adjacent

Here, angle = 65°, adjacent = 16 m (shadow), opposite = height

So, tan 65° = height / 16

Solve for height:

height = 16 × tan 65°

Now approximate:

height ≈ 16 × 2.1445 ≈ 34.4

Hence, the building is approximately 34.4 m tall.

Use sine with ramp length as hypotenuse: sin θ = opposite / hypotenuse

Here, opposite = 2.5 ft, hypotenuse = 12 ft

So, sin θ = 2.5 / 12 ≈ 0.2083

Now find the angle:

θ = arcsin(0.2083) ≈ 12°

Hence, the ramp’s angle of elevation is about 12°.

Use the tangent ratio: tan θ = rise / run

Here, rise = 4 m, run = 40 m

So, tan θ = 4 / 40 = 0.1

Now find the angle: θ = arctan(0.1) ≈ 5.7°

Hence, the bridge’s angle of elevation is about 5.7°.

Use sine to find the vertical height: height = hypotenuse × sin(angle)

So, substitute the values: height = 10 × sin 70°

Now approximate: height ≈ 10 × 0.9397 ≈ 9.4

Hence, the ladder reaches approximately 9.4 ft high.

Use sine to find the vertical height: height = hypotenuse × sin(angle)

So, substitute the values: height = 15 × sin 25°

Now approximate: height ≈ 15 × 0.4226 ≈ 6.3

Hence, the ramp’s height is approximately 6.3 ft.

Use the tangent ratio for angle of elevation: tan θ = opposite / adjacent

Here, opposite = 80 m (tower height)

adjacent = 55 m (shadow length)

So, tan θ = 80 / 55 ≈ 1.4545

Now find the angle: θ = arctan(1.4545) ≈ 56°

Hence, the sun’s angle of elevation is about 56°.

Use the tangent ratio: tan θ = opposite / adjacent

Here, opposite = 30 m (rise), adjacent = 200 m (horizontal distance)

So, tan θ = 30 / 200 = 0.15

Now find the angle: θ = arctan(0.15) ≈ 9°

Hence, the hill’s angle of elevation is about 9°.

Use sine to find the vertical height: height = hypotenuse × sin(angle)

Here, the guy wire is the hypotenuse: 20 m.

So, height = 20 × sin 60°

Use sin 60° = √3 / 2: height = 20 × (√3 / 2) = 10√3 ≈ 17.3

Hence, the pole is approximately 17.3 m tall.

Use sine to relate height and string length: sin(angle) = opposite / hypotenuse

Here, opposite = 50 m (height), angle = 35°, hypotenuse = string length

So, sin 35° = 50 / length

Solve for length: length = 50 / sin 35°

Now approximate: length ≈ 50 / 0.5736 ≈ 87.1

Hence, the string is approximately 87.1 m long.

Use tangent to relate height and shadow: tan(angle) = opposite / adjacent

Here, angle = 40°, adjacent = 22 m (shadow), opposite = height

So, tan 40° = height / 22

Solve for height:

height = 22 × tan 40°

Now approximate:

height ≈ 22 × 0.8391 ≈ 18.5

Hence, the tree’s height is approximately 18.5 m.

Use tangent for angle of elevation: tan θ = opposite / adjacent

Here, opposite = height of lighthouse, adjacent = 200 m

So, tan 20° = height / 200

Solve for height: height = 200 × tan 20°

Now approximate: height ≈ 200 × 0.36397 ≈ 72.8

Hence, the lighthouse is approximately 72.8 m tall.

Use sine with the pole as hypotenuse: sin θ = opposite / hypotenuse

Here, opposite = 28 ft, hypotenuse = 30 ft

So, sin θ = 28 / 30 ≈ 0.9333

Now find the angle: θ = arcsin(28/30) ≈ 69.7°

Use sine with slope distance as hypotenuse: sin θ = opposite / hypotenuse

Here, opposite = 120 m (vertical drop), hypotenuse = 400 m

So, sin θ = 120 / 400 = 0.3

Now find the angle: θ = arcsin(0.3) ≈ 17°

Hence, the angle of descent is about 17°.

Use sine with ladder as hypotenuse: sin θ = opposite / hypotenuse

Here, opposite = 30 ft, hypotenuse = 40 ft

So, sin θ = 30 / 40 = 0.75

Now find the angle: θ = arcsin(0.75) ≈ 48.6° ≈ 49°

Hence, the ladder makes about a 49° angle with the ground.

Use tangent for angle of elevation: tan θ = opposite / adjacent

Here, opposite = 50 m, adjacent = 60 m

So, tan θ = 50 / 60 ≈ 0.8333

Now find the angle: θ = arctan(0.8333) ≈ 40°

Hence, the sun’s angle of elevation is about 40°.

Use sine with cable as hypotenuse: sin θ = opposite / hypotenuse

Here, opposite = 100 ft, angle = 30°, hypotenuse = cable length

So, sin 30° = 100 / length= sin 30° = 1/2, so:1/2 = 100 / length

Solve for length:

length = 100 ÷ (1/2) = 200 ft

Hence, the cable is 200 ft long.

Use sine to find the vertical rise: rise = hypotenuse × sin(angle)

Here, hypotenuse = 25 m, angle = 10°

So, rise = 25 × sin 10°

Now approximate: rise ≈ 25 × 0.1736 ≈ 4.3

Hence, the ramp’s vertical rise is approximately 4.3 m.

Use tangent to relate height and horizontal distance: tan θ = opposite / adjacent

Here, opposite = 150 m (cliff height), angle = 25°, adjacent = distance from boat

So, tan 25° = 150 / distance

Solve for distance: distance = 150 / tan 25°

Now approximate: distance ≈ 150 / 0.4663 ≈ 321 m

Hence, the boat is approximately 321 m from the base of the cliff.

Use sine with roof length as hypotenuse: rise = hypotenuse × sin(angle)

Here, hypotenuse = 5 m, angle = 35°

So, rise = 5 × sin 35°

Now approximate: rise ≈ 5 × 0.5736 ≈ 2.9

Hence, the roof’s vertical rise is approximately 2.9 m.

Use tangent for the hill’s rise: rise = horizontal × tan(angle)

Here, horizontal = 150 m, angle = 12°

So, rise = 150 × tan 12°

Now approximate: rise ≈ 150 × 0.2126 ≈ 31.9 ≈ 32.0

Hence, the hill rises approximately 32.0 m over 150 m of horizontal run.