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

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

Options:

Correct Answer: 4/5

Solution:

Using the triangle with opposite = 3 and adjacent = 4, hypotenuse = 5. Thus, cos(tan^(-1)(3/4)) = 4/5.

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

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

Correct Answer: 4/5

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 follows: hypotenuse = sqrt(opposite^2 + adjacent^2) = sqrt(3^2 + 4^2) = sqrt(9 + 16) = sqrt(25) = 5.
Step 4: Now, we need to find the cosine of the angle. Cosine is defined as the adjacent side divided by the hypotenuse.
Step 5: Calculate cos(tan^(-1)(3/4)) = adjacent/hypotenuse = 4/5.

Inverse Trigonometric Functions – Understanding how to interpret and evaluate inverse trigonometric functions using right triangles.
Right Triangle Relationships – Applying the Pythagorean theorem to find the lengths of the sides of a right triangle.
Cosine Function – Evaluating the cosine of an angle based on the ratio of the adjacent side to the hypotenuse in a right triangle.