1/2 points details my notes larpcalclimaga8 5.4.043. 1/6 submissions used find the exact value of the…

Explanation:

Step1: Find $\cos(u)$

Since $\sin(u)=-\frac{3}{5}$ and $3\pi/2 < u<2\pi$, using $\sin^{2}u+\cos^{2}u = 1$, we have $\cos(u)=\sqrt{1-\sin^{2}(u)}=\sqrt{1 - (-\frac{3}{5})^{2}}=\sqrt{1-\frac{9}{25}}=\sqrt{\frac{16}{25}}=\frac{4}{5}$.

Step2: Find $\sin(v)$

Since $\cos(v)=\frac{15}{17}$ and $0 < v<\pi/2$, using $\sin^{2}v+\cos^{2}v = 1$, we have $\sin(v)=\sqrt{1-\cos^{2}(v)}=\sqrt{1 - (\frac{15}{17})^{2}}=\sqrt{1-\frac{225}{289}}=\sqrt{\frac{64}{289}}=\frac{8}{17}$.

Step3: Use the sum - formula for sine

The formula for $\sin(u + v)$ is $\sin(u + v)=\sin(u)\cos(v)+\cos(u)\sin(v)$. Substitute $\sin(u)=-\frac{3}{5}$, $\cos(u)=\frac{4}{5}$, $\cos(v)=\frac{15}{17}$, and $\sin(v)=\frac{8}{17}$ into the formula: $\sin(u + v)=(-\frac{3}{5})\times\frac{15}{17}+\frac{4}{5}\times\frac{8}{17}=-\frac{45}{85}+\frac{32}{85}=-\frac{13}{85}$.

Step4: Use the difference - formula for cosine

The formula for $\cos(u - v)$ is $\cos(u - v)=\cos(u)\cos(v)+\sin(u)\sin(v)$. Substitute $\sin(u)=-\frac{3}{5}$, $\cos(u)=\frac{4}{5}$, $\cos(v)=\frac{15}{17}$, and $\sin(v)=\frac{8}{17}$ into the formula: $\cos(u - v)=\frac{4}{5}\times\frac{15}{17}+(-\frac{3}{5})\times\frac{8}{17}=\frac{60}{85}-\frac{24}{85}=\frac{36}{85}$.

Answer:

For $\sin(u + v)$: $-\frac{13}{85}$ For $\cos(u - v)$: $\frac{36}{85}$