Question

question
in δnop, o = 48 inches, n = 47 inches and ∠n=72°. find all possible values of ∠o, to the nearest degree.
answer attempt 3 out of 3
additional angle no possible angles
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Explanation:

Step1: Apply the Law of Sines

In \(\triangle NOP\), by the Law of Sines, we have \(\frac{\sin\angle O}{n}=\frac{\sin\angle N}{o}\). We know \(o = 48\) inches, \(n = 47\) inches, and \(\angle N=72^\circ\). Plugging in the values, we get \(\frac{\sin\angle O}{47}=\frac{\sin72^\circ}{48}\).

Step2: Solve for \(\sin\angle O\)

First, calculate \(\sin72^\circ\approx0.9511\). Then, \(\sin\angle O=\frac{47\times\sin72^\circ}{48}\). Substituting the value of \(\sin72^\circ\), we have \(\sin\angle O=\frac{47\times0.9511}{48}\approx\frac{44.7017}{48}\approx0.9313\).

Step3: Find \(\angle O\)

Now, we find the angle whose sine is approximately \(0.9313\). Using the inverse sine function, \(\angle O=\sin^{-1}(0.9313)\approx68^\circ\). We also need to check if there is another possible angle. The supplement of \(68^\circ\) is \(180^\circ - 68^\circ=112^\circ\). Now we check if this angle is valid. The sum of angles in a triangle is \(180^\circ\). If \(\angle O = 112^\circ\) and \(\angle N = 72^\circ\), then \(\angle P=180^\circ-(112^\circ + 72^\circ)= - 4^\circ\), which is not possible. So only one possible angle exists.

Answer:

\(68^\circ\)