+118658 How can you simplify the equation y = sin(atan(x))?
atan of some number is between -90 and +90 degrees (by usual definition)
so sin of it can be positive or negative
Let's consider the case where the angle is an acute angle
Consider (draw) a right angled triangle triangle and put theta as an acute angle
opp side is x
adj side is 1
so atan(x)=theta
and hypotenuse is \(\sqrt{1+x^2}\)
so
\(sin\theta = \frac{opp}{hyp}\\ sin\theta = \frac{x}{1+x^2}\\ sin(atan(x)) =\pm \; \frac{x}{1+x^2}\\\)
How can you simplify the equation y = sin(atan(x))?
y = x/sqrt(x^2+1) and x^2+1 must not =0
Solve for x:
y = x/sqrt(x^2+1)
y = x/sqrt(x^2+1) is equivalent to x/sqrt(x^2+1) = y:
x/sqrt(x^2+1) = y
Multiply both sides by sqrt(x^2+1):
x = y sqrt(x^2+1)
x = sqrt(x^2+1) y is equivalent to sqrt(x^2+1) y = x:
y sqrt(x^2+1) = x
Raise both sides to the power of two:
y^2 (x^2+1) = x^2
Expand out terms of the left hand side:
y^2+x^2 y^2 = x^2
Subtract x^2+y^2 from both sides:
x^2 (y^2-1) = -y^2
Divide both sides by y^2-1:
x^2 = -y^2/(y^2-1)
Take the square root of both sides:
Answer: |x = i sqrt(y^2/(y^2 - 1)) or x = -i sqrt(y^2/(y^2 - 1))
