Answer:
[tex]cos(\alpha+\beta)=-\frac{84}{85}[/tex]
Step-by-step explanation:
we know that
[tex]cos(\alpha+\beta)=cos(\alpha)*cos(\beta)-sin(\alpha)*sin(\beta)[/tex]
Remember the identity
[tex]cos^{2} (x)+sin^2(x)=1[/tex]
step 1
Find the value of [tex]sin(\alpha)[/tex]
we have that
The angle alpha lie on the III Quadrant
so
The values of sine and cosine are negative
[tex]cos(\alpha)=-\frac{8}{17}[/tex]
Find the value of sine
[tex]cos^{2} (\alpha)+sin^2(\alpha)=1[/tex]
substitute
[tex](-\frac{8}{17})^{2}+sin^2(\alpha)=1[/tex]
[tex]sin^2(\alpha)=1-\frac{64}{289}[/tex]
[tex]sin^2(\alpha)=\frac{225}{289}[/tex]
[tex]sin(\alpha)=-\frac{15}{17}[/tex]
step 2
Find the value of [tex]cos(\beta)[/tex]
we have that
The angle beta lie on the IV Quadrant
so
The value of the cosine is positive and the value of the sine is negative
[tex]sin(\beta)=-\frac{4}{5}[/tex]
Find the value of cosine
[tex]cos^{2} (\beta)+sin^2(\beta)=1[/tex]
substitute
[tex](-\frac{4}{5})^{2}+cos^2(\beta)=1[/tex]
[tex]cos^2(\beta)=1-\frac{16}{25}[/tex]
[tex]cos^2(\beta)=\frac{9}{25}[/tex]
[tex]cos(\beta)=\frac{3}{5}[/tex]
step 3
Find cos (α + β)
[tex]cos(\alpha+\beta)=cos(\alpha)*cos(\beta)-sin(\alpha)*sin(\beta)[/tex]
we have
[tex]cos(\alpha)=-\frac{8}{17}[/tex]
[tex]sin(\alpha)=-\frac{15}{17}[/tex]
[tex]sin(\beta)=-\frac{4}{5}[/tex]
[tex]cos(\beta)=\frac{3}{5}[/tex]
substitute
[tex]cos(\alpha+\beta)=-\frac{8}{17}*\frac{3}{5}-(-\frac{15}{17})*(-\frac{4}{5})[/tex]
[tex]cos(\alpha+\beta)=-\frac{24}{85}-\frac{60}{85}[/tex]
[tex]cos(\alpha+\beta)=-\frac{84}{85}[/tex]
