Question 84221
The equation to use is:
{{{sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)}}}
You are given {{{sin(alpha)}}} and {{{sin(beta)}}}, so you need to get {{{cos(alpha)}}} and {{{cos(beta)}}} before you can solve the problem.
You can apply the Pythagorean theorem to see that the adjacent side of the triangle (x) to alpha = 1; Similarly, apply the Pythagorean theorem to see that the adjacent side of the triangle (y) to beta = 2:
{{{x^2+2^2=(sqrt(5))^2}}}
{{{x^2=5-4}}}
{{{x=1}}}
and:
{{{y^2+1^2=(sqrt(5))^2}}}
{{{y^2=5-1}}}
{{{y=2}}}
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Now you can figure out that
{{{cos(alpha)=1/sqrt(5)}}}
and
{{{cos(beta)=2/sqrt(5)}}}
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Now just plug and chug in the original equation:
{{{sin(alpha+beta)=sin(alpha)cos(beta)+cos(alpha)sin(beta)}}}
{{{sin(alpha+beta)=(2/sqrt(5))*(2/sqrt(5))+(1/sqrt(5))*(1/sqrt(5))}}}
{{{sin(alpha+beta)=4/5+1/5}}}
{{{highlight(sin(alpha+beta)=1)}}}
Good Luck, 
tutor_paul@yahoo.com