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Trigonometric functions How to determine a trigonometric function if i know tanx=m/n, is possible to find sinx ?

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Tushar Chandra eNotes educator | Certified Educator

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If you know that tan x = (m/n), you can find sin x.

Go about it this way: tan x = (m/n)

=> sin x/ cos x = m/n

now change cos x to sin x, use (cos x)^2 = 1 - (sin x)^2

=> sin x/ sqrt( 1- (sin x)^2) = m/n

=> (sin x)^2 / (1 - (sin x)^2) = m^2/n^2

=> (1 - (sin x)^2) / (sin x)^2 = n^2/m^2

=> 1/ (sin x)^2 - 1 = n^2/m^2

=> 1/ (sin x)^2 = n^2/m^2 +1 = (n^2 +m^2)/m^2

=> (sin x)^2 = m^2/ (n^2 + m^2)

=> sin x = sqrt [ m^2 / (n^2 + m^2)]

So sin x = m / sqrt (n^2 + m^2)

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giorgiana1976 | Student

Yes, it is possible. Do not be impressed by the parametric values of the numerator and denominator of tangent function.

We know that the tangent, in a right angle triangle, is the ratio between the opposite cathetus and the joined cathetus.

In this case, is given the tan x=m/n, so we'll conclude that one cathetus is m and the other one is n.

Also, in a right angle triangle,

sin x = opposite cathetus/hypotenuse=m/hypotenuse

But in a right angle triangle, by applying Pythagorean theorem:

(hypotenuse)^2=(cathetus)^2 + (cathetus)^2

(hypotenuse)^2 = m^2 + n^2

We'll substitute the formula above

sin x =p/hypotenuse

sin x = p/sqrt (m^2 + n^2)