use the given conditions to find the exact values of sin(2u), cos(2u), and tan(2u) using the double - angle…

Explanation:

Step1: Find cos(u)

Since $\sin(u)=-\frac{3}{5}$ and $\frac{3\pi}{2}<u < 2\pi$, we use $\sin^{2}u+\cos^{2}u = 1$. So $\cos(u)=\sqrt{1-\sin^{2}u}=\sqrt{1 - (-\frac{3}{5})^{2}}=\sqrt{1-\frac{9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}$ (because $u$ is in the fourth - quadrant where cosine is positive).

Step2: Find sin(2u)

Use the double - angle formula $\sin(2u)=2\sin(u)\cos(u)$. Substitute $\sin(u)=-\frac{3}{5}$ and $\cos(u)=\frac{4}{5}$ into it. Then $\sin(2u)=2\times(-\frac{3}{5})\times\frac{4}{5}=-\frac{24}{25}$.

Step3: Find cos(2u)

Use the double - angle formula $\cos(2u)=\cos^{2}u-\sin^{2}u$. Substitute $\sin(u)=-\frac{3}{5}$ and $\cos(u)=\frac{4}{5}$ into it. $\cos(2u)=(\frac{4}{5})^{2}-(-\frac{3}{5})^{2}=\frac{16}{25}-\frac{9}{25}=\frac{7}{25}$.

Step4: Find tan(2u)

Use the formula $\tan(2u)=\frac{\sin(2u)}{\cos(2u)}$. Substitute $\sin(2u)=-\frac{24}{25}$ and $\cos(2u)=\frac{7}{25}$ into it. Then $\tan(2u)=\frac{-\frac{24}{25}}{\frac{7}{25}}=-\frac{24}{7}$.

Answer:

$\sin(2u)=-\frac{24}{25}$ $\cos(2u)=\frac{7}{25}$ $\tan(2u)=-\frac{24}{7}$