Question

question 8 (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} \\) a) \\( \angle a = 43^{\circ} \\) b) \\( \angle a = 32^{\circ} \\) c) \\( \angle a = 51^{\circ} \\) d) \\( \angle a = 46^{\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 $B=81^{\circ}$ into the formula, we get $\frac{3.3}{\sin A}=\frac{6.2}{\sin 81^{\circ}}$.

Step2: Solve for $\sin A$

Cross - multiply: $6.2\sin A=3.3\sin81^{\circ}$.
Since $\sin81^{\circ}\approx0.9877$, then $6.2\sin A = 3.3\times0.9877=3.25941$.
So, $\sin A=\frac{3.25941}{6.2}\approx0.5257$.

Step3: Find the angle $A$

Using the inverse - sine function, $A=\sin^{-1}(0.5257)$.
$A\approx32^{\circ}$ (using a calculator to find the inverse sine value).

Answer:

B. $\angle A = 32^{\circ}$