Question

Problem 1) Suppose we design a triangular ramp with a base of 12 feet and an angle of 42°. What is the height of the ramp, DF? Use your trigonometric functions.



(1 point)

DF = 9.1

DF = 10.8

DF = 5.3

DF = 20.5
Problem 2) Suppose we design a triangular ramp with base = 10 feet and height = x. What is the height of the ramp, x? Use your trigonometric functions.



(1 point)

x = 8.4

x = 7.1

x = 5.6

x = 2.3
Problem 3) An engineer is designing a blueprint of a triangular ramp, where hypotenuse = 8cm and height = x. What is the height, x? Use your trigonometric functions.



(1 point)

x = 3.56

x = 2.39

x = 5.14

x = 10.21
Problem 4) An engineer is designing a ramp where hypotenuse = 14 and x = base. What is the length of the base, x? Use your trigonometric functions.



(1 point)

x = 12.24

x = 9.12

x = 1.14

x = 20.63
Problem 5) We are designing a triangular ramp that has the proportions of a 45-45-90 special right triangle. Based on the image below, what is the value of x for this ramp?



(1 point)

x = 9cm

x = 10cm

x = 11cm

x = 12cm
Problem 6) Engineers are building a triangular ramp to help cars ascend a steep cliff. The ramp has a hypotenuse of 4000m and rises at a 22° angle to the horizontal. What is the height, x, of the ramp?



(1 point)

x = 1005m

x = 909m

x = 1498m

x = 6001m
Problem 7) In your own words, explain what the phrase "SohCahToa" means. What do each of the letters stand for? What equations can we write with SohCahToa?(2 points)
0 / 10000 Word Limit
Problem 8) Please write about a real-world situation or problem that we could solve with the trigonometric functions (sine, cosine, and tangent). How could we use the trig functions to solve the problem? Be creative!

Hint: think about different real-world situations that involve right triangles.

(3 points)

Answer 1

Problem 1
- Height = opposite side = 12 * tan(42°) ≈ 12(0.9004) ≈ 10.8
Answer: DF = 10.8

Problem 2
- Height = opposite = 10 * tan(40°) ≈ 10(0.8391) ≈ 8.4
Answer: x = 8.4

Problem 3
- Height = hypotenuse * sin(angle). Using the angle shown (≈26.6°), x = 8 * sin(26.6°) ≈ 8(0.445) ≈ 3.56
Answer: x = 3.56

Problem 4
- Base = hypotenuse * cos(angle). Using the angle shown (≈29°), x = 14 * cos(29°) ≈ 14(0.8746) ≈ 12.24
Answer: x = 12.24

Problem 5
- 45-45-90 triangle: legs are equal and hypotenuse = leg * sqrt(2). If the given hypotenuse corresponds to a leg of 9, then x = 9 cm.
Answer: x = 9 cm

Problem 6
- Height = 4000 * sin(22°) ≈ 4000(0.3746) ≈ 1498 m
Answer: x = 1498 m

Problem 7 — What “SohCahToa” means
- “SohCahToa” is a memory aid for the basic trig ratios in a right triangle:
- SOH: sin(θ) = Opposite / Hypotenuse
- CAH: cos(θ) = Adjacent / Hypotenuse
- TOA: tan(θ) = Opposite / Adjacent
- From these you can also write useful relations, e.g. tan(θ) = sin(θ)/cos(θ), and the reciprocals: csc = 1/sin, sec = 1/cos, cot = 1/tan.

Problem 8 — Real-world situation using trig
Example: Measuring the height of a tree without climbing it.
- Procedure:
1. Stand a measured horizontal distance d from the tree (say d = 30 m).
2. Use a clinometer or smartphone app to measure the angle of elevation θ to the top of the tree (say θ = 35°).
3. The tree height (above your eye level) is h = d * tan(θ). So h = 30 * tan(35°) ≈ 30(0.7002) ≈ 21.0 m.
4. Add your eye height above ground to get the total tree height.
- Trig functions (here tangent) let you turn an easily measured distance and angle into an otherwise hard-to-measure height.