Questions on Geometry: Triangles answered by real tutors!

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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!: 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(1741) 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^2(x)+Cos^2(x)=1
Since Tan(x)=2, then
Opp/Adj = 2
Opp = 2 Adj
Opp/Hyp = 2 Adj/Hyp
Sin(x) = 2 Cos(x)
(2 Cos(x))^2+Cos^2(x)=1
4*Cos^2(x)+Cos^2(x)=1
5*Cos^2(x)=1
Cos^2(x)=1/5
Cos(x)=1/sqrt(5)
Cos(x)=.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,
(63.4-x)/(63.4-63.5)=(0.4478-0.4472)/(0.4478-0.4462)
(63.4-x)/(-0.1)=(0.0006)/(0.0016)
63.4-x=-.0375
x=63.44
From the calculator,
x=63.43