Question

use the following diagram to answer question 2.

1. use the diagram above to solve for the unknown sides. round to one decimal place.

a. find side ( k ).
we know
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b. find side ( m ).
we know
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Explanation:

Step1: Find side \( k \) using the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}\). In the given triangle, we know one side \(45\) cm and its opposite angle \(20^{\circ}- 10^{\circ}=10^{\circ}\), and we want to find \(k\) whose opposite angle is \(90^{\circ}-10^{\circ} = 80^{\circ}\).
So, \(\frac{45}{\sin10^{\circ}}=\frac{k}{\sin80^{\circ}}\).
Cross - multiplying gives \(k=\frac{45\times\sin80^{\circ}}{\sin10^{\circ}}\).
Since \(\sin80^{\circ}\approx0.9848\) and \(\sin10^{\circ}\approx0.1736\), then \(k=\frac{45\times0.9848}{0.1736}\approx255.5\) cm.

Step2: Find side \( m \) using the Law of Sines

We know from the Law of Sines \(\frac{45}{\sin10^{\circ}}=\frac{m}{\sin(90^{\circ}-20^{\circ})}\).
Since \(\sin(90^{\circ}-20^{\circ})=\cos20^{\circ}\approx0.9397\), then \(m = \frac{45\times\cos20^{\circ}}{\sin10^{\circ}}\).
Substituting the values \(\sin10^{\circ}\approx0.1736\) and \(\cos20^{\circ}\approx0.9397\), we get \(m=\frac{45\times0.9397}{0.1736}\approx244.0\) cm.

Answer:

a. \(k\approx255.5\) cm
b. \(m\approx244.0\) cm