SOLUTION: if alpha and beta are acute angles with sin alpha=2/sqrt5 and beta=1/sqrt5, find exact value of sin(alpha+beta)

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Question 84221: if alpha and beta are acute angles with sin alpha=2/sqrt5 and beta=1/sqrt5, find exact value of sin(alpha+beta)
Answer by tutor_paul(519) About Me  (Show Source):
You can put this solution on YOUR website!
The equation to use is:
sin%28alpha%2Bbeta%29=sin%28alpha%29cos%28beta%29%2Bcos%28alpha%29sin%28beta%29
You are given sin%28alpha%29 and sin%28beta%29, so you need to get cos%28alpha%29 and cos%28beta%29 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%5E2%2B2%5E2=%28sqrt%285%29%29%5E2
x%5E2=5-4
x=1
and:
y%5E2%2B1%5E2=%28sqrt%285%29%29%5E2
y%5E2=5-1
y=2
===========================
Now you can figure out that
cos%28alpha%29=1%2Fsqrt%285%29
and
cos%28beta%29=2%2Fsqrt%285%29
===========================
Now just plug and chug in the original equation:
sin%28alpha%2Bbeta%29=sin%28alpha%29cos%28beta%29%2Bcos%28alpha%29sin%28beta%29

sin%28alpha%2Bbeta%29=4%2F5%2B1%2F5
highlight%28sin%28alpha%2Bbeta%29=1%29
Good Luck,
tutor_paul@yahoo.com