Question

find all missing elements.
a = 5
b = 2
c = 6
a = ?° b = ° c = °
round to the nearest tenth.

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 \( a = 5 \), \( b = 2 \), \( c = 6 \) into the formula:
\( 5^{2}=2^{2}+6^{2}-2\times2\times6\times\cos A \)
\( 25=4 + 36-24\cos A \)
\( 25=40-24\cos A \)
\( 24\cos A=40 - 25 \)
\( 24\cos A=15 \)
\( \cos A=\frac{15}{24}=\frac{5}{8}=0.625 \)
\( A=\cos^{-1}(0.625)\approx51.3^{\circ} \)

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

The Law of Cosines formula is \( b^{2}=a^{2}+c^{2}-2ac\cos B \).
Substitute \( a = 5 \), \( b = 2 \), \( c = 6 \) into the formula:
\( 2^{2}=5^{2}+6^{2}-2\times5\times6\times\cos B \)
\( 4=25 + 36-60\cos B \)
\( 4=61-60\cos B \)
\( 60\cos B=61 - 4 \)
\( 60\cos B=57 \)
\( \cos B=\frac{57}{60}=0.95 \)
\( B=\cos^{-1}(0.95)\approx18.2^{\circ} \)

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

Since the sum of angles in a triangle is \( 180^{\circ} \), \( C=180^{\circ}-A - B \)
Substitute \( A\approx51.3^{\circ} \) and \( B\approx18.2^{\circ} \)
\( C=180-(51.3 + 18.2)=110.5^{\circ} \)

Answer:

\( A = 51.3^{\circ} \), \( B = 18.2^{\circ} \), \( C = 110.5^{\circ} \)