Answer:
sin(α + β) = -84/205
tan(α + β) = 84/187
Explanation:
To find sin(α + β), we will use the following identity
[tex]\begin{gathered} \sin ^2\theta+\cos ^2\theta=1 \\ \text{Then} \\ \sin ^2(\alpha+\beta)^{}+\cos ^2(\alpha+\beta)^{}=1 \end{gathered}[/tex]So, solving for sin(α + β), we get:
[tex]\sin (\alpha+\beta)=\pm_{}\sqrt[]{1-\cos^2(\alpha+\beta)}[/tex]Now, we can replace cos(α + β) = -187/205 to get:
[tex]\begin{gathered} \sin (\alpha+\beta)=\pm\sqrt[]{1-(\frac{187}{205})^2} \\ \sin (\alpha+\beta)=\pm\frac{84}{205} \end{gathered}[/tex]Then, α + β is on quadrant III. It means that the sine of the angle is negative. Therefore
sin(α + β) = -84/205
Finally, to add tan(α + β), we will use the following
[tex]\tan (\alpha+\beta)=\frac{\sin (\alpha+\beta)}{cos(\alpha+\beta)}[/tex]Replacing the values, we get:
[tex]\tan (\alpha+\beta)=\frac{-\frac{84}{205}}{-\frac{187}{205}}=\frac{-84\times205}{205\times(-187)}=\frac{84}{187}[/tex]Therefore
tan(α + β) = 84/187