Question

Write the following as an algebraic expression in u, u > 0. sin (artan u/ square root 3)​

Answer 1

Answer: Let's start by using the identity:

tan(arctan(x)) = x

to simplify the expression inside the sine function. So, we have:

arctan(u) / sqrt(3) = tan(arctan(u) / sqrt(3))

Now, using the trigonometric identity:

tan(x/y) = sin(x) / (cos(y) + sin(y))

with x = arctan(u) and y = sqrt(3), we get:

tan(arctan(u) / sqrt(3)) = sin(arctan(u)) / (cos(sqrt(3)) + sin(sqrt(3)))

Simplifying further, we know that:

sin(arctan(u)) = u / sqrt(1 + u^2)

and

cos(sqrt(3)) + sin(sqrt(3)) = 2cos(sqrt(3) - pi/4)

So, the expression becomes:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2cos(sqrt(3) - pi/4)]

Simplifying the denominator, we have:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / (2(cos(sqrt(3))cos(pi/4) + sin(sqrt(3))sin(pi/4)))

Using the values for cosine and sine of pi/4, we get:

cos(pi/4) = sin(pi/4) = 1/sqrt(2)

So, we have:

sin(arctan(u) / sqrt(3)) = u / sqrt(1 + u^2) / [2(sqrt(3)/2 + 1/2)]

Simplifying further:

sin(arctan(u) / sqrt(3)) = u / (sqrt(1 + u^2) * (sqrt(3) + 1))

Therefore, the algebraic expression for sin(arctan(u) / sqrt(3)) is:

u / (sqrt(1 + u^2) * (sqrt(3) + 1))

Explanation: