Question

question
in \\( \triangle ijk \\), \\( i = 100 \\) inches, \\( m\angle i = 115^\circ \\) and \\( m\angle j = 23^\circ \\). find the length of \\( j \\), to the nearest inch.
answer attempt 1 out of 3
\\( j = \\) \\( \square \\) inches
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Explanation:

Step1: Find angle K

In a triangle, the sum of angles is \(180^\circ\). So \(m\angle K = 180^\circ - m\angle I - m\angle J\).
Substituting values: \(m\angle K = 180 - 115 - 23 = 42^\circ\).

Step2: Apply Law of Sines

Law of Sines: \(\frac{i}{\sin I} = \frac{j}{\sin J}\).
We know \(i = 100\), \(m\angle I = 115^\circ\), \(m\angle J = 23^\circ\).
Rearrange for \(j\): \(j = \frac{i \cdot \sin J}{\sin I}\).

Step3: Calculate sine values

\(\sin 115^\circ \approx \sin(65^\circ) \approx 0.9063\) (since \(\sin(180 - x) = \sin x\)), \(\sin 23^\circ \approx 0.3907\).

Step4: Compute \(j\)

Substitute into formula: \(j = \frac{100 \cdot 0.3907}{0.9063} \approx \frac{39.07}{0.9063} \approx 43.1\).
Round to nearest inch: \(j \approx 43\).

Answer:

43