Question

how do we use law to find a side?
example 1: in, given that ∠b = 48°, ∠c = 25°, and side a (named as bc) bc = 36 cm. find the length of ab and ac correct to 1 decimal place.

Explanation:

Step1: Find angle \(A\)

The sum of angles in a triangle is \(180^{\circ}\). So, \(A = 180^{\circ}-(48^{\circ}+25^{\circ})=107^{\circ}\)

Step2: Use the Law of Sines for side \(AB\) (side \(c\))

The Law of Sines is \(\frac{a}{\sin A}=\frac{c}{\sin C}\). Substituting \(a = 36\), \(A = 107^{\circ}\), \(C=25^{\circ}\), we get \(c=\frac{36\times\sin25^{\circ}}{\sin107^{\circ}}\).
\(\sin25^{\circ}\approx0.4226\), \(\sin107^{\circ}\approx0.9563\)
\(c=\frac{36\times0.4226}{0.9563}=\frac{15.2136}{0.9563}\approx15.9\)

Step3: Use the Law of Sines for side \(AC\) (side \(b\))

Using \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Substituting \(a = 36\), \(A = 107^{\circ}\), \(B = 48^{\circ}\)
\(\sin48^{\circ}\approx0.7431\), \(\sin107^{\circ}\approx0.9563\)
\(b=\frac{36\times\sin48^{\circ}}{\sin107^{\circ}}=\frac{36\times0.7431}{0.9563}=\frac{26.7516}{0.9563}\approx28.0\)

Answer:

\(AB\approx15.9\) cm and \(AC\approx28.0\) cm