Answer:
x ≈ 11.5
Step-by-step explanation:
To find the length ( x ) in a triangle using the Law of Cosines, we can follow these steps:
Given:
[tex](a = 12) \\ (b = 11)\\ ( gamma = 60^circ )[/tex]
The Law of Cosines states:
[tex]\[ c^2 = a^2 + b^2 - 2ab \cos(\gamma) \][/tex]
Here, ( x ) corresponds to ( c ). Thus,
[tex]\[ x^2 = 12^2 + 11^2 - 2 \times 12 \times 11 \times \cos(60^\circ) \]
[/tex]
First, compute the cosine of
[tex]( 60^ \circ ):
\cos(60^\circ) = 0.5 \]
[/tex]
Substitute the values into the formula:
[tex]\[ x^2 = 12^2 + 11^2 - 2 \times 12 \times 11 \times 0.5 \][/tex]
Calculate each term step-by-step:
[tex]\[ x^2 = 144 + 121 - 2 \times 12 \times 11 \times 0.5 \][/tex]
[tex][ x^2 = 144 + 121 - 132 ][/tex]
[tex]
\[ x^2 = 265 - 132 \][/tex]
[tex]\[ x^2 = 133 \][/tex]
Take the square root to find ( x ):
[tex]\[ x = \sqrt{133} \]
[/tex]
[tex]\[ x \approx 11.5 \][/tex]
So, the length ( x ) is approximately ( 11.5 ) when rounded to one decimal place.
