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Identify the known sides relative to angle alpha.

Mathematics

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2 parts
a

α=30∘\alpha = 30^\circ

θ=60∘\theta = 60^\circ

b

232\sqrt{3}

Step-by-step solution

8 steps

Here's the solution for question 4:

Part a· 4 steps

  1. Identify the known sides relative to angle α\alpha.

    The side opposite to angle α\alpha is AC=2AC = 2.
    The hypotenuse is AB=4AB = 4.

  2. Use the sine ratio to find angle α\alpha.

    sin⁡(α)=OppositeHypotenuse=ACAB=24=12\sin(\alpha) = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AC}{AB} = \frac{2}{4} = \frac{1}{2}

  3. Calculate α\alpha.

    α=arcsin⁡(12)\alpha = \arcsin\left(\frac{1}{2}\right) α=30∘\alpha = 30^\circ

  4. Use the sum of angles in a triangle to find angle θ\theta.

    The sum of angles in a triangle is 180∘180^\circ, and ∠C=90∘\angle C = 90^\circ.

    θ+α+90∘=180∘\theta + \alpha + 90^\circ = 180^\circ θ+30∘+90∘=180∘\theta + 30^\circ + 90^\circ = 180^\circ θ+120∘=180∘\theta + 120^\circ = 180^\circ θ=180∘−120∘\theta = 180^\circ - 120^\circ θ=60∘\theta = 60^\circ

    The values of the angles are α=30∘\boxed{\alpha = 30^\circ} and θ=60∘\boxed{\theta = 60^\circ}.

Part b

232\sqrt{3}

  1. Use the Pythagorean theorem to find the length of side BCBC.

    AC2+BC2=AB2AC^2 + BC^2 = AB^2

  2. Substitute the known values AC=2AC=2 and AB=4AB=4.

    22+BC2=422^2 + BC^2 = 4^2 4+BC2=164 + BC^2 = 16

  3. Solve for BC2BC^2.

    BC2=16−4BC^2 = 16 - 4 BC2=12BC^2 = 12

  4. Take the square root and simplify the surd.

    BC=12BC = \sqrt{12} BC=4×3BC = \sqrt{4 \times 3} BC=4×3BC = \sqrt{4} \times \sqrt{3} BC=23BC = 2\sqrt{3}

    The length of side BCBC is 23\boxed{2\sqrt{3}}.

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