Question

question
in δmno, n = 370 inches, m = 810 inches and ∠m=21°. find all possible values of ∠n, to the nearest 10th of a degree.
answer attempt 3 out of 3
additional angle no possible angles
submit answer
watch video
show examples

Explanation:

Step1: Apply the Law of Sines

In \(\triangle MNO\), by the Law of Sines, we have \(\frac{\sin N}{n}=\frac{\sin M}{m}\). Substituting the given values \(n = 370\), \(m = 810\), and \(\angle M=21^{\circ}\), we get \(\frac{\sin N}{370}=\frac{\sin 21^{\circ}}{810}\).

Step2: Solve for \(\sin N\)

First, calculate \(\sin 21^{\circ}\approx0.3584\). Then, \(\sin N=\frac{370\times\sin 21^{\circ}}{810}=\frac{370\times0.3584}{810}\approx\frac{132.608}{810}\approx0.1637\).

Step3: Find the possible angles for \(N\)

We know that if \(\sin\theta = x\), then \(\theta=\arcsin(x)\) or \(\theta = 180^{\circ}-\arcsin(x)\) (within \(0^{\circ}<\theta<180^{\circ}\) for a triangle angle).

• Calculate \(\arcsin(0.1637)\approx9.4^{\circ}\).
• Calculate \(180^{\circ}-9.4^{\circ}=170.6^{\circ}\). But we need to check if this angle is valid. The sum of angles in a triangle is \(180^{\circ}\). If \(N = 170.6^{\circ}\) and \(M = 21^{\circ}\), then \(O=180^{\circ}-170.6^{\circ}-21^{\circ}=-11.6^{\circ}\), which is not possible. So we discard \(170.6^{\circ}\).

Answer:

\(9.4^{\circ}\)