Question

find the length of side a.
c = 35° a =? b =
round to the nearest tenth.

Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In \(\triangle ABC\), we know \(B = 100^{\circ}\), \(C=35^{\circ}\), so \(A=180^{\circ}-100^{\circ}-35^{\circ}=45^{\circ}\), and \(BC = a\), \(AC = 15\).
We use the formula \(\frac{a}{\sin A}=\frac{AC}{\sin B}\).

Step2: Substitute values into the formula

Substitute \(A = 45^{\circ}\), \(AC = 15\), \(B = 100^{\circ}\) into \(\frac{a}{\sin A}=\frac{AC}{\sin B}\).
We get \(a=\frac{15\times\sin45^{\circ}}{\sin100^{\circ}}\).
Since \(\sin45^{\circ}=\frac{\sqrt{2}}{2}\approx0.707\), \(\sin100^{\circ}\approx0.985\).
Then \(a=\frac{15\times0.707}{0.985}\).

Step3: Calculate the value of \(a\)

\(15\times0.707 = 10.605\), and \(\frac{10.605}{0.985}\approx10.8\).

Answer:

\(10.8\)