We know that sin(arctan x) can be expressed in terms of x by using the relationship between trigonometric functions and inverse trigonometric functions.
Specifically, we can use the identities:
1. tan(arctan x) = x (definition of arctan)
2. sin^2(arctan x) + cos^2(arctan x) = 1 (Pythagorean identity)
From the first identity, we know that tan(arctan x) = x. By taking the reciprocal of this equation, we get that cos(arctan x) = 1 / sqrt(1 + x^2).
Now, we can use the Pythagorean identity to find sin^2(arctan x):
sin^2(arctan x) + cos^2(arctan x) = 1
sin^2(arctan x) + (1 / sqrt(1 + x^2))^2 = 1
sin^2(arctan x) + 1 / (1 + x^2) = 1
sin^2(arctan x) = 1 - 1 / (1 + x^2)
sin^2(arctan x) = (1 + x^2) / (1 + x^2) - 1 / (1 + x^2)
sin^2(arctan x) = x^2 / (1 + x^2)
Taking the square root of both sides, we have:
sin(arctan x) = sqrt(x^2 / (1 + x^2)) = x / sqrt(1 + x^2)
Therefore, sin(arctan x) in terms of x is x / sqrt(1 + x^2).
sin(arctan x) in terms of x. explain how you know
1 answer