Question

find the length of side c.

law of cosines: ( c^2 = a^2 + b^2 - 2ab cdot cos(c) )
round your answer to the nearest hundredth.

Explanation:

Step1: Identify given values

We have \( a = 9 \), \( b = 12 \), and \( C = 102^\circ \). The Law of Cosines formula is \( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) \).

Step2: Substitute values into formula

First, calculate \( a^2 \), \( b^2 \), and \( \cos(102^\circ) \).
\( a^2 = 9^2 = 81 \), \( b^2 = 12^2 = 144 \).
\( \cos(102^\circ) \approx \cos(102) \approx -0.2079 \) (using a calculator).
Now substitute into the formula:
\( c^2 = 81 + 144 - 2 \times 9 \times 12 \times (-0.2079) \)

Step3: Simplify the expression

Calculate each part:
\( 81 + 144 = 225 \)
\( 2 \times 9 \times 12 = 216 \)
\( 216 \times (-0.2079) = -44.9064 \), but since it's \( -2ab \cos(C) \), it becomes \( -216 \times (-0.2079) = 44.9064 \)
So \( c^2 = 225 + 44.9064 = 269.9064 \)

Step4: Take the square root

\( c = \sqrt{269.9064} \approx 16.43 \) (rounded to the nearest hundredth)

Answer:

\( 16.43 \)