3. if cos(x) = 3/5 and sin x < 0, find sin(2x) as a fraction in simplest terms. sin(2x) = ______ -24/25 4/5…

Explanation:

Step1: Find sin(x) using identity

We know that $\sin^{2}x+\cos^{2}x = 1$. Given $\cos(x)=\frac{3}{5}$, then $\sin^{2}x=1 - \cos^{2}x$. Substituting $\cos(x)=\frac{3}{5}$, we get $\sin^{2}x=1-(\frac{3}{5})^{2}=1-\frac{9}{25}=\frac{16}{25}$. Since $\sin(x)<0$, then $\sin(x)=-\frac{4}{5}$.

Step2: Use double - angle formula

The double - angle formula for sine is $\sin(2x) = 2\sin(x)\cos(x)$. Substitute $\sin(x)=-\frac{4}{5}$ and $\cos(x)=\frac{3}{5}$ into the formula. So $\sin(2x)=2\times(-\frac{4}{5})\times\frac{3}{5}=-\frac{24}{25}$.

Answer:

A. $-\frac{24}{25}$