To determine the value of sin (θ + β), we can apply the trigonometric identity:
[tex]\sin (\theta+\beta)=\sin \theta\cos \beta+\cos \theta\sin \beta[/tex]
Since we already have the value for cosθ, let's find out sin θ.
Based on trigonometric identity,
[tex]\sin \theta=\frac{y}{r};\cos \theta=\frac{x}{r}[/tex]
Based on the given value of cosθ, x = -√2 while r = 3. To determine the value of y, let's apply the Pythagorean Theorem.
[tex]\begin{gathered} y=\sqrt[]{r^2-x^2} \\ y=\sqrt[]{3^2-(-\sqrt[]{2})^2} \\ y=\sqrt[]{9-2} \\ y=\sqrt[]{7} \end{gathered}[/tex]
Since the given interval is between π and 3π/2 which is in Quadrant 3, y = -√7. Hence, the value of sin θ is:
[tex]\sin \theta=-\frac{\sqrt[]{7}}{3}[/tex]
The next thing that we shall solve is sin β and cos β. We can use the given tangent function to determine this.
[tex]\begin{gathered} \text{tan}\beta=\frac{y}{x}=\frac{4}{3} \\ \text{Solve for r.} \\ r=\sqrt[]{x^2+y^2}=\sqrt[]{3^2+4^2}=\sqrt[]{9+16}=\sqrt[]{25}=5 \end{gathered}[/tex]
Given the interval for beta, the angle is found in Quadrant 1. So, the values of sin β and cos β are:
[tex]\begin{gathered} \sin \beta=\frac{y}{r}=\frac{4}{5} \\ \cos \beta=\frac{x}{r}=\frac{3}{5} \end{gathered}[/tex]
Now that we have the values for sin θ = -√7/3, cos β = 3/5, cos θ = -√2/3, and sin β = 4/5, let's plugged them into the first trigonometric identity we mentioned above.
[tex]\begin{gathered} \sin (\theta+\beta)=\sin \theta\cos \beta+\cos \theta\sin \beta \\ \sin (\theta+\beta)=(-\frac{\sqrt[]{7}}{3})(\frac{3}{5})+(-\frac{\sqrt[]{2}}{3})(\frac{4}{5}) \end{gathered}[/tex]
Then, simplify.
[tex]\sin (\theta+\beta)=-\frac{\sqrt[]{7}}{5}-\frac{4\sqrt[]{2}}{15}[/tex]
The final answer is shown above.
