Find the value of sin(tan^(-1)(x)).

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Question: Find the value of sin(tan^(-1)(x)).

Options:

Correct Answer: x/√(1+x^2)

Solution:

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

Find the value of sin(tan^(-1)(x)).

Find the value of sin(tan^(-1)(x)).

Correct Answer: x/√(1+x^2)

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 (θ) such that tan(θ) = x.
Step 3: In a right triangle, tangent is defined as opposite/adjacent. So, if we let the opposite side be x and the adjacent side be 1, we have tan(θ) = x/1.
Step 4: Use the Pythagorean theorem to find the hypotenuse. The hypotenuse (h) is calculated as h = √(opposite^2 + adjacent^2) = √(x^2 + 1^2) = √(x^2 + 1).
Step 5: Now, we need to find sin(θ). The sine of an angle is defined as opposite/hypotenuse. So, sin(θ) = opposite/hypotenuse = x/√(x^2 + 1).
Step 6: Therefore, sin(tan^(-1)(x)) = x/√(1 + x^2).

Inverse Trigonometric Functions – Understanding how to interpret and manipulate inverse trigonometric functions, specifically tan^(-1)(x), in the context of right triangles.
Right Triangle Relationships – Applying the definitions of sine, cosine, and tangent in a right triangle to derive relationships between the sides.
Pythagorean Theorem – Using the Pythagorean theorem to find the hypotenuse when given one side of a right triangle.