Question

b) determine the measure of $\theta$ to the nearest degree.
37.1 cm
29.5 cm
44°

Explanation:

Step1: Apply the sine - law

The sine - law states that $\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$. In $\triangle ABC$, let $a = 37.1$ cm, $b = 29.5$ cm, and $B = 44^{\circ}$. We want to find $\theta$ (angle $C$). First, we use the formula $\frac{a}{\sin A}=\frac{b}{\sin B}$.
$\frac{37.1}{\sin A}=\frac{29.5}{\sin44^{\circ}}$

Step2: Solve for $\sin A$

Cross - multiply to get $\sin A=\frac{37.1\times\sin44^{\circ}}{29.5}$.
$\sin A=\frac{37.1\times0.6947}{29.5}=\frac{25.7734}{29.5}\approx0.8737$
$A=\sin^{- 1}(0.8737)\approx61^{\circ}$

Step3: Find angle $\theta$ (angle $C$)

Since the sum of the interior angles of a triangle is $180^{\circ}$, $A + B+\theta=180^{\circ}$.
$\theta = 180^{\circ}-A - B$. Substitute $A\approx61^{\circ}$ and $B = 44^{\circ}$ into the formula.
$\theta=180^{\circ}-61^{\circ}-44^{\circ}=75^{\circ}$

Answer:

$75^{\circ}$