Math Problem Statement

Calculate the bearing of point C from point A. B is 5 km due east of turbine A, and C is 8 km due north of turbine B. Give your answer correct to 1 decimal place.

Solution

We are tasked with calculating the bearing of point CC from point AA.

Step-by-step approach:

  1. Identify the right triangle: The diagram shows a right-angled triangle ABC\triangle ABC with:

    • AB=5kmAB = 5 \, \text{km} (the horizontal distance)
    • BC=8kmBC = 8 \, \text{km} (the vertical distance).
  2. Bearing definition: A bearing is measured clockwise from north. We need to find the angle θ\theta, where θ\theta is the angle between line ACAC and the north direction from AA.

  3. Trigonometric relationship:

    • The bearing angle θ\theta can be found using the tangent function, since we have the opposite side BCBC and the adjacent side ABAB: tan(θ)=ABBC=58.\tan(\theta) = \frac{AB}{BC} = \frac{5}{8}.
  4. Calculate the angle: θ=tan1(58).\theta = \tan^{-1}\left(\frac{5}{8}\right).

Let me calculate this for you.The bearing of point CC from point AA is 58.058.0^\circ, measured clockwise from north.

Would you like more details or have any questions?

Here are some related questions to expand on this concept:

  1. How do bearings change when distances are reversed between points?
  2. How can you use other trigonometric functions to find angles in similar problems?
  3. What is the significance of the right triangle in navigation problems like this one?
  4. How would the bearing change if point CC was directly east of point AA?
  5. How can Pythagoras' Theorem be used to check the distances in this problem?

Tip: Bearings are always measured clockwise from the north, and a full bearing can range from 00^\circ to 360360^\circ.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Bearings
Right-Angled Triangle

Formulas

tan(θ) = opposite / adjacent
θ = tan⁻¹(opposite/adjacent)

Theorems

Pythagorean Theorem
Trigonometric Ratios

Suitable Grade Level

Grades 10-12