SOLUTION: what is the sine of an angle whose tangent is 2? first find an answer without a calculator, then check it with one. thank you!

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Question 148692: what is the sine of an angle whose tangent is 2? first find an answer without a calculator, then check it with one.
thank you!
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
From trigonometry, you know the following identities, where
Opp = Length of triangle leg opposite to angle x.
Adj = Length of triangle leg adjacent to angle x.
Hyp = Hypotenuse of triangle with sides Opp and Adj.
Tan(x)=Opp/Adj
Sin(x)=Opp/Hyp
Cos(x)=Adj/Hyp
Sin%5E2%28x%29%2BCos%5E2%28x%29=1
Since Tan(x)=2, then
Opp/Adj = 2
Opp = 2 Adj
Opp/Hyp = 2 Adj/Hyp
Sin(x) = 2 Cos(x)
%282+Cos%28x%29%29%5E2%2BCos%5E2%28x%29=1
4%2ACos%5E2%28x%29%2BCos%5E2%28x%29=1
5%2ACos%5E2%28x%29=1
Cos%5E2%28x%29=1%2F5
Cos%28x%29=1%2Fsqrt%285%29
Cos%28x%29=.4472
Checking the sine tables,
0.4478 for 63.4
0.4462 for 63.5
We can set up a proportion to get an approximate answer,
%2863.4-x%29%2F%2863.4-63.5%29=%280.4478-0.4472%29%2F%280.4478-0.4462%29
%2863.4-x%29%2F%28-0.1%29=%280.0006%29%2F%280.0016%29
63.4-x=-.0375
x=63.44
From the calculator,
x=63.43