Question

question 9 (mandatory) (1 point)
determine \\( \angle a \\) to the nearest degree for the triangle with the given information.
\\( a = 3.3 \mathrm { m }, b = 6.2 \mathrm { m }, \angle b = 81 ^ { \circ } \\)
\\( \bigcirc \\) a) \\( \angle a = 51 ^ { \circ } \\)
\\( \bigcirc \\) b) \\( \angle a = 46 ^ { \circ } \\)
\\( \bigcirc \\) c) \\( \angle a = 43 ^ { \circ } \\)
\\( \bigcirc \\) d) \\( \angle a = 32 ^ { \circ } \\)

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Substituting the given values \(a = 3.3\), \(b = 6.2\), and \(B=81^{\circ}\), we get \(\frac{3.3}{\sin A}=\frac{6.2}{\sin 81^{\circ}}\).

Step2: Solve for \(\sin A\)

Cross - multiply: \(6.2\sin A=3.3\sin 81^{\circ}\).
Since \(\sin 81^{\circ}\approx0.9877\), then \(6.2\sin A = 3.3\times0.9877\).
\(6.2\sin A=3.25941\).
\(\sin A=\frac{3.25941}{6.2}\).
\(\sin A\approx0.5257\).

Step3: Find \(\angle A\)

Take the inverse - sine: \(A=\sin^{-1}(0.5257)\).
Using a calculator, \(A\approx32^{\circ}\).

Answer:

d) \(\angle A = 32^{\circ}\)