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 θ. So, tan(θ) = 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 = √(3^2 + 1^2) = √(9 + 1) = √10.
Step 5: Now we can find cos(θ). Cosine is defined as the adjacent side over the hypotenuse. So, cos(θ) = adjacent/hypotenuse = 1/√10.
Step 6: Therefore, cos(tan^(-1)(3)) = 1/√10.
Trigonometric Functions and Inverses – 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 – Using the properties of right triangles to derive values of trigonometric functions based on given ratios.
