Sure, let’s solve for angle [tex]\( C \)[/tex] in degrees.
We are given the equation:
[tex]\[ 9^2 = 4^2 + 7^2 - 2(4)(7) \cos C \][/tex]
First, simplify each term:
[tex]\[ 81 = 16 + 49 - 56 \cos C \][/tex]
Combine the constants on the right side:
[tex]\[ 81 = 65 - 56 \cos C \][/tex]
Rearrange the equation to isolate the cosine term:
[tex]\[ 81 - 65 = -56 \cos C \][/tex]
[tex]\[ 16 = -56 \cos C \][/tex]
Solve for [tex]\( \cos C \)[/tex]:
[tex]\[ \cos C = \frac{16}{-56} \][/tex]
[tex]\[ \cos C = -0.2857142857142857 \][/tex]
To find angle [tex]\( C \)[/tex], we use the inverse cosine function ([tex]\(\cos^{-1}\)[/tex]):
[tex]\[ C = \cos^{-1}(-0.2857142857142857) \][/tex]
This gives us [tex]\( C \)[/tex] in radians:
[tex]\[ C \approx 1.8605480282309441 \text{ radians} \][/tex]
Convert the angle from radians to degrees:
[tex]\[ C \approx 1.8605480282309441 \times \frac{180}{\pi} \][/tex]
[tex]\[ C \approx 106.60154959902025 \text{ degrees} \][/tex]
Round this to the nearest tenth:
[tex]\[ C \approx 106.6 \text{ degrees} \][/tex]
Therefore, the measure of angle [tex]\( C \)[/tex] in degrees, rounded to the nearest tenth, is:
[tex]\[ \boxed{106.6} \][/tex]
