Question

BlackTom AI · Accepted Answer

The problem describes triangle ABC with angles labeled as follows: ∠BAC = 3θ at A, and ∠ABC = θ at B, leaving ∠ACB = 180° − 4θ by the triangle angle sum.

We are given side lengths: AC = 4 m and BC = 8 m. Using the Law of Sines in triangle ABC, we relate side lengths to opposite angles:
- Side BC (length 8) is opposite angle ∠A = 3θ.
- Side AC (length 4) is opposite angle ∠B = θ.
Thus, 8 / sin(3θ) = 4 / sin(θ).

From this equation, cross-multiply to obtain 8 sin θ = 4 sin(3θ). Simplify by dividing both sides by 4: 2 sin θ = sin(3θ).

Recall the triple-angle expansion: sin(3θ) = 3 sin θ − 4 sin^3 θ. Substitute this into the equation:
2 sin θ = 3 sin θ − 4 sin^3 θ.

Bring all terms to one side: 0 = sin θ − 4 sin^3 θ = sin θ (1 − 4 sin^2 θ).

For a nondegenerate triangle, sin θ ≠ 0, so we must have 1 − 4 sin^2 θ = 0, which gives sin^2 θ = 1/4 and hence sin θ = 1/2 (taking the positive root since θ is an angle in a triangle).

Therefore, θ is the angle with sine 1/2, which gives θ = 30° (within the usual range 0° < θ < 180° in a triangle).

Note on the provided options: The prompt lists an answer as 30°, but no multiple choices are shown in the given data. The reasoning above derives θ = 30° from the given side lengths and angle relationships using the Law of Sines and basic trigonometric identities.

CIVL1810 (ND) CIVL9810 (ND) Early Feedback Task

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