Question

in \\( \triangle ghi, h = 770 \\) inches, \\( i = 870 \\) inches and \\( \angle g = 9 ^ { \circ } \\). find the length of \\( g \\), to the nearest inch.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \(g^{2}=h^{2}+i^{2}-2hi\cos G\).
Substitute \(h = 770\), \(i = 870\), and \(G = 9^{\circ}\) into the formula:
\(g^{2}=770^{2}+870^{2}-2\times770\times870\times\cos(9^{\circ})\)

Step2: Calculate each term

• Calculate \(770^{2}=770\times770 = 592900\)
• Calculate \(870^{2}=870\times870=756900\)
• Calculate \(2\times770\times870 = 2\times669900=1339800\)
• Calculate \(\cos(9^{\circ})\approx0.9877\)

Then \(2\times770\times870\times\cos(9^{\circ})\approx1339800\times0.9877 = 1323320.46\)

Step3: Compute \(g^{2}\)

\(g^{2}=592900 + 756900-1323320.46\)
\(g^{2}=1349800-1323320.46\)
\(g^{2}=26479.54\)

Step4: Find \(g\)

Take the square root of \(g^{2}\): \(g=\sqrt{26479.54}\approx163\)

Answer:

\(163\) inches