Question

How would I go about solving this? x+y)" align="absmiddle" class="latex-formula"> if cosx=8/17 and siny=12/37

Answer 1

$\bf cos(x)=\cfrac{8}{17}\cfrac{\leftarrow adjacent=a}{\leftarrow hypotenuse=c} \\\\\\ \textit{using the pythagorean theorem} \\\\\\ c^2=a^2+b^2\implies \sqrt{c^2-a^2}=b\implies \sqrt{17^2-8^2}=b\implies 15=b\\\\ -----------------------------\\\\ sin(y)=\cfrac{12}{37}\cfrac{\leftarrow opposite=b}{\leftarrow hypotenuse=c}\\\\\\ \textit{using the pythagorean theorem} \\\\\\ c^2=a^2+b^2\implies\sqrt{c^2-b^2}=a\implies \sqrt{37^2-12^2}=a\implies 35=a$

$\bf \\\\ -----------------------------\\\\ sin({{ \alpha}} + {{ \beta}})=sin({{ \alpha}})cos({{ \beta}}) + cos({{ \alpha}})sin({{ \beta}})\qquad thus \\\\\\ sin(x+y)=sin(x)cos(y)+cos(x)sin(y) \\\\\\ sin(x+y)=\cfrac{15}{17}\cdot \cfrac{35}{37}+\cfrac{8}{17}\cdot \cfrac{12}{37} \\\\\\ sin(x+y)=\cfrac{525}{629}+\cfrac{96}{629}\implies sin(x+y)=\cfrac{525+96}{629} \\\\\\ sin(x+y)=\cfrac{621}{629}$