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

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

Options:

Correct Answer: √2/2

Solution:

sin(tan^(-1)(1)) = 1/√2, using the triangle with opposite and adjacent sides equal.

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

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

Correct Answer: 1/√2

Step 1: Understand that tan^(-1)(1) means we are looking for an angle whose tangent is 1.
Step 2: Recall that tangent is the ratio of the opposite side to the adjacent side in a right triangle.
Step 3: The angle whose tangent is 1 is 45 degrees (or π/4 radians), because the opposite and adjacent sides are equal.
Step 4: In a right triangle with a 45-degree angle, both the opposite and adjacent sides can be considered equal, let's say each side is 1.
Step 5: Use the Pythagorean theorem to find the hypotenuse: hypotenuse = √(1^2 + 1^2) = √2.
Step 6: Now, find the sine of the angle: sin(45 degrees) = opposite/hypotenuse = 1/√2.
Step 7: Therefore, sin(tan^(-1)(1)) = 1/√2.

Inverse Trigonometric Functions – Understanding how to evaluate the sine of an angle derived from the inverse tangent function.
Right Triangle Relationships – Using the properties of right triangles to find the sine value based on the sides of the triangle.