Question

solve for the remaining angles and side of the one triangle that can be created. round to the nearest hundredth:

b = 45°, b = 6.5, a = 6

Explanation:

Step1: Use the Law of Sines to find angle \(A\)

The Law of Sines is \(\frac{a}{\sin A}=\frac{b}{\sin B}\).
Substitute \(a = 6\), \(b = 6.5\), and \(B=45^{\circ}\) into the formula: \(\sin A=\frac{a\sin B}{b}\).

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\(A=\sin^{- 1}(0.6526)\approx40.74^{\circ}\)

Step2: Find angle \(C\)

Since the sum of angles in a triangle is \(180^{\circ}\), \(C = 180^{\circ}-A - B\).
Substitute \(A\approx40.74^{\circ}\) and \(B = 45^{\circ}\): \(C=180^{\circ}-40.74^{\circ}-45^{\circ}=94.26^{\circ}\)

Step3: Use the Law of Sines to find side \(c\)

Again, using the Law of Sines \(\frac{c}{\sin C}=\frac{b}{\sin B}\).
\(c=\frac{b\sin C}{\sin B}\)
Substitute \(b = 6.5\), \(B = 45^{\circ}\), and \(C = 94.26^{\circ}\)

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Answer:

\(A\approx40.74^{\circ}\), \(C\approx94.26^{\circ}\), \(c\approx9.17\)