Question

question
in \\( \triangle wxy \\), \\( w = 670 \mathrm { cm } , y = 800 \mathrm { cm } \\) and \\( \angle y = 84 ^ { \circ } \\). find all possible values of \\( \angle w \\), to the nearest 10th of a degree.
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attempt 1 out of 2
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Explanation:

Step1: Apply the Law of Sines

The Law of Sines states that \(\frac{\sin W}{w}=\frac{\sin Y}{y}\). Substituting the given values \(w = 670\), \(y=800\), and \(\angle Y = 84^{\circ}\), we get \(\frac{\sin W}{670}=\frac{\sin84^{\circ}}{800}\).

Step2: Solve for \(\sin W\)

Cross - multiply: \(\sin W=\frac{670\times\sin84^{\circ}}{800}\). Calculate \(\sin84^{\circ}\approx0.9945\). Then \(\sin W=\frac{670\times0.9945}{800}=\frac{666.315}{800}\approx0.8329\).

Step3: Find the acute angle \(W\)

Using the inverse sine function, \(W_1=\sin^{- 1}(0.8329)\approx56.4^{\circ}\).

Step4: Find the obtuse angle \(W\)

Since \(\sin\theta=\sin(180^{\circ}-\theta)\), the obtuse angle \(W_2 = 180^{\circ}-56.4^{\circ}=123.6^{\circ}\). Check the triangle angle sum: For \(W_1 = 56.4^{\circ}\), the third angle \(Z_1=180-(56.4 + 84)=39.6^{\circ}\). For \(W_2 = 123.6^{\circ}\), the third angle \(Z_2=180-(123.6 + 84)= - 27.6^{\circ}\) (invalid).

Answer:

\(56.4^{\circ}\)