The value of sin(tan^(-1)(x)) is:

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Question: The value of sin(tan^(-1)(x)) is:

Options:

Correct Answer: x/√(1+x^2)

Solution:

Using the right triangle definition, sin(tan^(-1)(x)) = opposite/hypotenuse = x/√(1+x^2).

The value of sin(tan^(-1)(x)) is:

The value of sin(tan^(-1)(x)) is:

Step 1: Understand that tan^(-1)(x) is the angle whose tangent is x.
Step 2: Draw a right triangle where the angle is theta, and the opposite side is x and the adjacent side is 1 (since tan(theta) = opposite/adjacent = x/1).
Step 3: Use the Pythagorean theorem to find the hypotenuse. The hypotenuse is √(opposite^2 + adjacent^2) = √(x^2 + 1^2) = √(x^2 + 1).
Step 4: Now, find sin(theta). Sin is defined as opposite/hypotenuse, so sin(tan^(-1)(x)) = x / √(x^2 + 1).
Step 5: Simplify the expression to get sin(tan^(-1)(x)) = x / √(1 + x^2).

Trigonometric Functions – Understanding the relationships between sine, tangent, and the sides of a right triangle.
Inverse Trigonometric Functions – Using inverse functions to find angles and their corresponding trigonometric values.
Pythagorean Theorem – Applying the theorem to relate the sides of a right triangle.