Question

solve for the remaining sides and angles if possible (if not, answer \dne\ in all answer boxes). as in the text. (a,a), (b,b), and (c,c) are angle - side opposite pairs. round to one decimal places, if necessary. a = 11, b = 12, c = 13

Explanation:

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

The Law of Cosines formula is \(a^{2}=b^{2}+c^{2}-2bc\cos A\).
Substitute \(a = 11\), \(b = 12\), \(c = 13\) into the formula:
\(11^{2}=12^{2}+13^{2}-2\times12\times13\times\cos A\)
\(121=144 + 169-312\cos A\)
\(121=313-312\cos A\)
\(312\cos A=313 - 121\)
\(312\cos A=192\)
\(\cos A=\frac{192}{312}\approx0.615\)
\(A=\cos^{-1}(0.615)\approx52.1^{\circ}\)

Step2: Use the Law of Cosines to find angle \(B\)

Using the Law of Cosines formula \(b^{2}=a^{2}+c^{2}-2ac\cos B\)
Substitute \(a = 11\), \(b = 12\), \(c = 13\)
\(12^{2}=11^{2}+13^{2}-2\times11\times13\times\cos B\)
\(144=121+169 - 286\cos B\)
\(144=290-286\cos B\)
\(286\cos B=290 - 144\)
\(286\cos B=146\)
\(\cos B=\frac{146}{286}\approx0.510\)
\(B=\cos^{-1}(0.510)\approx59.3^{\circ}\)

Step3: Use the angle - sum property of a triangle to find angle \(C\)

Since \(A + B + C=180^{\circ}\)
\(C=180-(A + B)\)
\(C=180-(52.1 + 59.3)\)
\(C = 68.6^{\circ}\)

Answer:

\(A\approx52.1^{\circ}\), \(B\approx59.3^{\circ}\), \(C\approx68.6^{\circ}\)