Use the given sum or difference identity to find sin⁡(α+β)\displaystyle {\sin{{\left(\alpha+\beta\right)}}}  exactly, given that sin⁡α=34\displaystyle {\sin{\alpha}}=\frac{{3}}{{4}} such that α\displaystyle \alpha terminates in quadrant II and cos⁡β=−14\displaystyle {\cos{\beta}}=-\frac{{1}}{{4}} such that β\displaystyle \beta terminates in quadrant III then:

 sin⁡(α+β)=sin⁡(α)cos⁡(β)+cos⁡(α)sin⁡(β)\displaystyle {\sin{{\left(\alpha+\beta\right)}}}={\sin{{\left(\alpha\right)}}}{\cos{{\left(\beta\right)}}}+{\cos{{\left(\alpha\right)}}}{\sin{{\left(\beta\right)}}} 

 sin⁡(α+β)=(34)(−14)+\displaystyle {\sin{{\left(\alpha+\beta\right)}}}={\left(\frac{{3}}{{4}}\right)}{\left(-\frac{{1}}{{4}}\right)}+(\displaystyle {(} )(\displaystyle {)}{(} )\displaystyle {)} 

 sin⁡(α+β)=\displaystyle {\sin{{\left(\alpha+\beta\right)}}}=

 

Your answers should be exact but you do not have to rationalize your denominators.

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