Question

applying the law of cosines with sas
which equation correctly uses the law of cosines to solve for the length s?
$9^2 = s^2 + 10^2 - 2(s)(10)\cos(100^\circ)$
$9 = s + 10 - 2(s)(10)\cos(100^\circ)$
$s^2 = 9^2 + 10^2 - 2(9)(10)\cos(100^\circ)$

Explanation:

Step1: Recall Law of Cosines

The Law of Cosines states that for a triangle with sides \(a\), \(b\), and \(c\), and the angle \(C\) opposite side \(c\), the formula is \(c^{2}=a^{2}+b^{2}-2ab\cos(C)\).

Step2: Identify sides and angle

In triangle \(UST\), we want to find side \(s\) (opposite the \(100^{\circ}\) angle at \(S\)). The other two sides are \(9\) and \(10\). So, let \(a = 9\), \(b = 10\), \(c = s\), and \(C=100^{\circ}\).

Step3: Apply Law of Cosines

Substitute into the formula: \(s^{2}=9^{2}+10^{2}-2(9)(10)\cos(100^{\circ})\).

Answer:

\(s^{2}=9^{2}+10^{2}-2(9)(10)\cos(100^{\circ})\) (the third equation)