Find the exact value of the following expression without using a calculator: tan(arcsin(x/2)).

To find the exact value of the expression tan(arcsin(x/2)), we need to understand that arcsin, which is the inverse of the sine function, gives us an angle whose sine value is a certain value within its domain, [-1, 1]. The expression arcsin(x/2) can be interpreted as the angle (let's call it θ) for which sin(θ) = x/2.

To find tan(θ), we can use the right triangle definition of the trigonometric functions. Suppose we have a right triangle where the angle θ is one of the non-right angles. The other non-right angle would be (90° - θ) or (π/2 - θ) in radians. According to the definition of the sine function for a right triangle, sin(θ) = Opposite / Hypotenuse.

Let's set: - Opposite = x (the numerator of the ratio x/2) - Hypotenuse = 2 (the denominator of the ratio x/2)

For a right triangle, we can use the Pythagorean theorem to find the length of the third side, which is the Adjacent side in the context of angle θ. The Pythagorean theorem states that for a right triangle with sides a, b, and hypotenuse c: a² + b² = c².

We'll call the Adjacent side 'a.' So we have: a² + x² = 2² a² + x² = 4

Solving for a: a² = 4 - x² a = √(4 - x²) (keeping in mind that 'a' must be positive since it represents a length)

Now, we know that the tangent function is tan(θ) = Opposite / Adjacent, so we can write: tan(θ) = x / √(4 - x²)

Thus, tan(arcsin(x/2)) = x / √(4 - x²). This is the exact value of the expression tan(arcsin(x/2)) without using a calculator.