Final answer:
The cosine of the arctangent of -1/3 is calculated using a right-angled triangle and is found to be 3√10/10, after rationalizing the denominator.
Step-by-step explanation:
To evaluate cos(tan-1(-(1)/(3))), we consider a right-angled triangle where the opposite side is -1 and the adjacent side is 3. We can find the hypotenuse of this triangle using the Pythagorean theorem, which will be √(12 + 32) = √10. The negative sign indicates that the angle is in the second or fourth quadrant, but since the range of tan-1 is (-π/2, π/2), the angle must be in the fourth quadrant. The cosine of an angle in the fourth quadrant is positive.
The exact value of cos(tan-1(-(1)/(3))) is therefore the adjacent side divided by the hypotenuse, which is 3/√10. Normalizing the denominator by rationalizing it, we get 3√10/10.