Question

given the triangle

, find the length of side ( x ) using the law of sines. round your final answer to 4 decimal places.

( x = )

question help:
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Explanation:

Step1: Find the third angle

The sum of angles in a triangle is \(180^{\circ}\). Let the third angle be \(A\). Then \(A = 180-(41 + 17)=122^{\circ}\).

Step2: Apply the Law of Sines

The Law of Sines states that \(\frac{a}{\sin A}=\frac{b}{\sin B}\). Here, if we assume the side of length \(17\) is opposite the \(122^{\circ}\) angle and \(x\) is opposite the \(17^{\circ}\) angle. So \(\frac{17}{\sin122^{\circ}}=\frac{x}{\sin17^{\circ}}\).

Step3: Solve for \(x\)

We know that \(\sin122^{\circ}=\sin(180 - 58)^{\circ}=\sin58^{\circ}\approx0.8480\) and \(\sin17^{\circ}\approx0.2924\). Then \(x=\frac{17\times\sin17^{\circ}}{\sin122^{\circ}}\). Substitute the values: \(x=\frac{17\times0.2924}{0.8480}\).

$$x=\frac{4.9708}{0.8480}\approx5.8618$$

Answer:

\(5.8618\)