Write the following as an algebraic expression in u, u>0.
cos(arctan(u/sqrt2))
tangent of theta = u/sqrt 2 = opposite/adjacent
let u = 1
then v = sqrt 2 = adjacent leg
and hypotenuse = sqrt (1 + 2) = sqrt 3
then cos theta = sqrt(2/3)
But Damon, that would only be the case when u = 1
e.g. when u = 4, we have tanØ = 4/√2
r^2 = 16 + 2 = 18
r = √18
then cosØ = √2/√18 = 1/3
in general case
r^2 = 2 + u^2
r = √(2 + u^2)
cos(arctan(u/√2)) = √2/√(2+u^2)
To write the expression cos(arctan(u/sqrt2)) as an algebraic expression, we can use the trigonometric identities.
The identity we will use is:
cos(arctan(x)) = sqrt(1 / (1 + x^2))
In our case, x = u / sqrt(2).
Now, we can substitute x into the identity:
cos(arctan(u/sqrt2)) = sqrt(1 / (1 + (u/sqrt2)^2))
To simplify this expression further, let's simplify the denominator:
cos(arctan(u/sqrt2)) = sqrt(1 / (1 + u^2/2))
Now, we can simplify by multiplying the numerator and denominator by 2 to get rid of the fraction in the denominator:
cos(arctan(u/sqrt2)) = sqrt(2 / (2 + u^2))
Hence, the expression cos(arctan(u/sqrt2)) can be written as sqrt(2 / (2 + u^2)).
To express cos(arctan(u/sqrt2)) as an algebraic expression in u, we'll use the identity cos(arctan(x)) = 1 / sqrt(1 + x^2).
First, let's find x in arctan(u/sqrt2).
The arctan function gives us the angle whose tangent is u/sqrt2, so we have:
tan(arctan(u/sqrt2)) = u/sqrt2.
Since the tangent function is the ratio of the opposite side to the adjacent side in a right triangle, we can construct a right triangle. Let the length of the opposite side be u and the length of the adjacent side be sqrt2. The hypotenuse can be found using the Pythagorean theorem:
(u)^2 + (sqrt2)^2 = hypotenuse^2
u^2 + 2 = hypotenuse^2
hypotenuse = sqrt(u^2 + 2)
So, in terms of x, we have x = u/sqrt(u^2 + 2).
Now, we can substitute this value of x into the identity for cos(arctan(x)):
cos(arctan(u/sqrt2)) = 1 / sqrt(1 + x^2)
cos(arctan(u/sqrt2)) = 1 / sqrt(1 + (u/sqrt(u^2 + 2))^2)
Therefore, the algebraic expression for cos(arctan(u/sqrt2)) is:
1 / sqrt(1 + (u/sqrt(u^2 + 2))^2)