Step 1: Understand that tan^(-1)(3) means we are looking for an angle whose tangent is 3.
Step 2: Let's call this angle ΞΈ. So, tan(ΞΈ) = 3.
Step 3: We can represent this tangent in terms of a right triangle. If the opposite side is 3 and the adjacent side is 1, we have a right triangle.
Step 4: Use the Pythagorean theorem to find the hypotenuse. The hypotenuse = β(3^2 + 1^2) = β(9 + 1) = β10.
Step 5: Now we can find cos(ΞΈ). Cosine is defined as the adjacent side over the hypotenuse. So, cos(ΞΈ) = adjacent/hypotenuse = 1/β10.
Step 6: Therefore, cos(tan^(-1)(3)) = 1/β10.
Trigonometric Functions and Inverses β Understanding the relationship between trigonometric functions and their inverses, specifically how to evaluate cosine of an angle given by the inverse tangent.
Right Triangle Relationships β Using the properties of right triangles to derive values of trigonometric functions based on given ratios.
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