Evaluate sin(tan^(-1)(3/4)).

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Question: Evaluate sin(tan^(-1)(3/4)).

Options:

Correct Answer: 3/5

Solution:

Using the right triangle definition, sin(tan^(-1)(3/4)) = opposite/hypotenuse = 3/5.

Evaluate sin(tan^(-1)(3/4)).

Evaluate sin(tan^(-1)(3/4)).

Correct Answer: 3/5

Step 1: Understand that tan^(-1)(3/4) means we are looking for an angle whose tangent is 3/4.
Step 2: Draw a right triangle where the opposite side is 3 and the adjacent side is 4.
Step 3: Use the Pythagorean theorem to find the hypotenuse. Calculate it as: hypotenuse = sqrt(opposite^2 + adjacent^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
Step 4: Now, find the sine of the angle. Sine is defined as opposite/hypotenuse.
Step 5: Substitute the values: sin(tan^(-1)(3/4)) = opposite/hypotenuse = 3/5.

Inverse Trigonometric Functions – Understanding how to evaluate the sine of an angle given by the inverse tangent function.
Right Triangle Relationships – Applying the definitions of sine, cosine, and tangent in the context of a right triangle.