Give an algebraic expression for the following:
π in(arctan π₯)
Thank you!!
x/β(x^2+1)
Draw the triangle to see why.
To find the algebraic expression for π in(arctan π₯), we can use the basic trigonometric identities.
First, let's recall the definition of sine and arctangent functions. The sine of an angle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse in a right triangle. The arctangent of a value is the angle whose tangent is that value.
Now, let's consider a right triangle with an angle π, opposite side of length π₯, and adjacent side of length 1. Using the definition of tangent, we have tan(π) = π₯/1 = π₯.
Taking the inverse tangent of both sides, we get arctan(tan(π)) = arctan(π₯).
Since sine is the ratio of the opposite side to the hypotenuse, we can now express π in(arctan π₯) as sin(arctan(π₯)) = π₯/β(1 + π₯^2).
Therefore, the algebraic expression for π in(arctan π₯) is π₯/β(1 + π₯^2).