Question

what is the length of side q, given r = 20, s = 30, and q = 15°? round the answer to the nearest tenth.

Explanation:

Step1: Apply the Law of Cosines

The Law of Cosines formula is \( q^{2}=r^{2}+s^{2}-2rs\cos Q \).
Substitute \( r = 20\), \(s = 30\), and \(Q=15^{\circ}\) into the formula.
So, \( q^{2}=20^{2}+30^{2}-2\times20\times30\times\cos(15^{\circ})\).

Step2: Calculate each term

First, \(20^{2}=400\), \(30^{2}=900\).
Next, \(\cos(15^{\circ})=\cos(45^{\circ}-30^{\circ})=\cos45^{\circ}\cos30^{\circ}+\sin45^{\circ}\sin30^{\circ}=\frac{\sqrt{2}}{2}\times\frac{\sqrt{3}}{2}+\frac{\sqrt{2}}{2}\times\frac{1}{2}=\frac{\sqrt{6}+\sqrt{2}}{4}\approx0.9659\).
Then, \(2\times20\times30\times\cos(15^{\circ}) = 1200\times0.9659 = 1159.08\).
And \(20^{2}+30^{2}=400 + 900=1300\).
So, \( q^{2}=1300-1159.08 = 140.92\).

Step3: Find \(q\)

Take the square root of \(q^{2}\), \(q=\sqrt{140.92}\approx11.9\).

Answer:

\(11.9\)