Question

solve for a.
(image of triangle abc with angle at c: 68°, angle at a: 83°, side c (ab) = 13, side a (bc) is to be found. a = ?)

Explanation:

Step1: Find angle B

In a triangle, the sum of angles is \(180^\circ\). So, \(\angle B = 180^\circ - 68^\circ - 83^\circ = 29^\circ\).

Step2: Apply the Law of Sines

The Law of Sines states \(\frac{a}{\sin A}=\frac{c}{\sin C}\). Here, \(A = 83^\circ\), \(c = 13\), \(C = 68^\circ\). So, \(\frac{a}{\sin 83^\circ}=\frac{13}{\sin 68^\circ}\).

Step3: Solve for a

Rearrange the formula: \(a=\frac{13\times\sin 83^\circ}{\sin 68^\circ}\). Calculate \(\sin 83^\circ\approx0.9925\), \(\sin 68^\circ\approx0.9272\). Then \(a=\frac{13\times0.9925}{0.9272}\approx\frac{12.9025}{0.9272}\approx13.92\).

Answer:

\(a\approx13.92\) (or more precise value depending on calculation precision)