Question

right triangle relationships and trigonometry
what is the measure of \\( \angle n \\) to the nearest whole degree?
\\( 65 ^ { \circ } \\)
\\( 30 ^ { \circ } \\)
\\( 66 ^ { \circ } \\)
\\( 45 ^ { \circ } \\)

Explanation:

Step1: Recall the tangent function

In a right - triangle, \(\tan(N)=\frac{\text{opposite}}{\text{adjacent}}\). Here, the side opposite to \(\angle N\) is \(LM = 5.74\) and the side adjacent to \(\angle N\) is \(LN=4\). So, \(\tan(N)=\frac{5.74}{4}=1.435\).

Step2: Find the angle using the inverse tangent function

We know that if \(\tan(N) = x\), then \(N=\tan^{- 1}(x)\). So, \(N=\tan^{-1}(1.435)\). Using a calculator, \(N\approx55^{\circ}\) (since \(\tan^{-1}(1.435)\approx55^{\circ}\) when rounded to the nearest whole degree).

Answer:

\(55^{\circ}\)