Question

find the length of side a.
a 18° 100 c
b 28° a c
a = ? c = □° c = □

Explanation:

Step1: Find angle C

The sum of angles in a triangle is \(180^\circ\). So, \( \angle C = 180^\circ - 18^\circ - 28^\circ = 134^\circ \)

Step2: Apply the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). Here, side \( b = 100 \) (opposite angle \( B = 28^\circ \)), angle \( A = 18^\circ \), and we want to find \( a \) (opposite angle \( A \)). So, \( \frac{a}{\sin 18^\circ} = \frac{100}{\sin 28^\circ} \)

Step3: Solve for \( a \)

\( a = \frac{100 \times \sin 18^\circ}{\sin 28^\circ} \)
Calculate \( \sin 18^\circ \approx 0.3090 \) and \( \sin 28^\circ \approx 0.4695 \)
\( a = \frac{100 \times 0.3090}{0.4695} \approx \frac{30.90}{0.4695} \approx 65.8 \) (rounded to one decimal place)

Answer:

\( a \approx 65.8 \) (If more precision is needed, use more decimal places for sine values. Also, angle \( C = 134^\circ \), and for \( c \), using Law of Sines \( c = \frac{100 \times \sin 134^\circ}{\sin 28^\circ} \), \( \sin 134^\circ \approx 0.6947 \), so \( c \approx \frac{100 \times 0.6947}{0.4695} \approx 147.9 \))