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 know \(a = 7\), \(b = 8\), and \(\angle C=32^\circ\). We use the Law of Cosines formula \(c^{2}=a^{2}+b^{2}-2ab\cdot\cos(C)\).

Step2: Substitute values into formula

Substitute \(a = 7\), \(b = 8\), and \(\cos(32^\circ)\approx0.8480\) into the formula:

$$
LATEXBLOCK0
$$

Step3: Calculate each part

First, calculate \(112\times0.8480 = 94.976\). Then, \(113-94.976 = 18.024\). So \(c^{2}=18.024\).

Step4: Find \(c\)

Take the square root of \(18.024\): \(c=\sqrt{18.024}\approx4.245\approx4.25\) (rounded to the nearest hundredth).

Answer:

\(4.25\)