From the statement, we know that:
[tex]\begin{gathered} \cos \theta=-\frac{\sqrt[]{2}}{3},\text{ where }\pi\leq\theta\leq\frac{3}{2}\pi, \\ \tan \beta=\frac{4}{3},\text{ where 0}\leq\theta\leq\frac{\pi}{2}\text{.} \end{gathered}[/tex]
1) First, we look for the value of angle θ.
Plotting both sides of the equation, we have the graph:
Using a calculator, we get the following value of θ:
[tex]\theta^{\prime}=\arccos (-\frac{\sqrt[]{2}}{3})\cong2.06.[/tex]
But to get an angle in the interval π ≤ θ ≤3π/2, the correct result is:
[tex]\theta=2\pi-\theta^{\prime}\cong4.22.[/tex]
We can check this result from the graph above.
The of θ is:
[tex]\sin (\theta)=-\frac{\sqrt[]{7}}{3}\text{.}[/tex]
2) Secondly, we look for the value of angle β.
Using a calculator, we get the following value of β:
[tex]\beta=\arctan (\frac{4}{3})\cong0.93.[/tex]
This result is in interval 0 ≤ θ ≤ π/2.
The sine and cosine of β are:
[tex]\begin{gathered} \sin \beta=\frac{4}{5}, \\ \cos \beta=\frac{3}{5}. \end{gathered}[/tex]
3) Using the results above, the sine of the sum of the angles θ and β is:
[tex]\sin (\theta+\beta)=\sin \theta\cdot\cos \beta+\sin \beta\cdot\cos \theta\text{.}[/tex]
Replacing the values obtained above, we get:
[tex]\sin (\theta+\beta)=(-\frac{\sqrt[]{7}}{3})\cdot\frac{3}{5}+\frac{4}{5}\cdot(-\frac{\sqrt[]{2}}{3})=-\frac{\sqrt[]{7}}{5}-\frac{4\cdot\sqrt[]{2}}{15}\text{.}[/tex]
Answer
[tex]\sin (\theta+\beta)=-\frac{\sqrt[]{7}}{5}-\frac{4\cdot\sqrt[]{2}}{15}[/tex]
