Evaluate sin(tan^(-1)(x)).

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Question: Evaluate sin(tan^(-1)(x)).

Options:

Correct Answer: x/√(1+x^2)

Solution:

Using the identity, sin(tan^(-1)(x)) = x/√(1+x^2).

Evaluate sin(tan^(-1)(x)).

Evaluate sin(tan^(-1)(x)).

Step 1: Understand that tan^(-1)(x) is the angle whose tangent is x.
Step 2: Let θ = tan^(-1)(x). This means tan(θ) = x.
Step 3: Recall the definition of tangent: tan(θ) = opposite/adjacent. Here, we can think of opposite = x and adjacent = 1.
Step 4: Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(opposite^2 + adjacent^2) = √(x^2 + 1).
Step 5: Now, we can find sin(θ). Since sin(θ) = opposite/hypotenuse, we have sin(θ) = x/√(x^2 + 1).
Step 6: Therefore, sin(tan^(-1)(x)) = x/√(1 + x^2).

Inverse Trigonometric Functions – Understanding how to evaluate the sine of an angle given in terms of the inverse tangent function.
Pythagorean Identity – Applying the relationship between sine, cosine, and tangent to derive the result.