Write the composed trigonometric function sin(arctan x) in terms of x. Explain your steps and/or show your work. Remember to rationalize the denominator if necessary.
aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
To find the composed trigonometric function sin(arctan x), we can start by considering a right triangle.
Let's assume we have a right triangle with one of its acute angles denoted as θ, and the opposite side is x and the adjacent side is 1.
By definition, the tangent of θ is equal to the ratio of the opposite side to the adjacent side, which gives us:
tan(θ) = x/1 = x
Now, we can find the value of θ by taking the inverse tangent (arctan) of both sides:
θ = arctan(x)
Since we are interested in finding sin(arctan x), we need to find the value of sin(θ). In order to do that, we can use the Pythagorean Theorem.
According to the Pythagorean Theorem, in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides.
So in our triangle, the hypotenuse is given by:
hypotenuse² = adjacent side² + opposite side²
hypotenuse² = 1² + x²
hypotenuse² = 1 + x²
Taking the square root of both sides, we get:
hypotenuse = √(1 + x²)
Now, we can find sin(θ) by dividing the opposite side (x) by the hypotenuse (√(1 + x²)):
sin(θ) = x/√(1 + x²)
In order to rationalize the denominator, we can multiply the numerator and denominator by √(1 + x²):
sin(θ) = (x * √(1 + x²))/[(√(1 + x²)) * √(1 + x²)]
sin(θ) = (x * √(1 + x²))/(√((1 + x²) * (1 + x²)))
sin(θ) = (x * √(1 + x²))/(√((1 + x²)²))
sin(θ) = (x * √(1 + x²))/(1 + x²)
Therefore, the composed trigonometric function sin(arctan x) can be expressed as (x * √(1 + x²))/(1 + x²).
same
I will just do one, then you try
draw right triangle A B C, C is right angle. angle at A is arc tan x
let side AC =1
side BC = x
side AC = sqrt(x^2 + 1)
sin A = x/sqrt(x^2+1)
= x sqrt(x^2+1) / ( x^2+1)