Final answer:
The exact value of cot(-π/3) is found by taking the reciprocal of tan(-π/3). Since the tangent of -π/3 is √3, the cotangent is the reciprocal, which is √3/3 after rationalizing the denominator.
Step-by-step explanation:
To find the exact value of the trigonometric function cotangent at the given real number -π/3, we first recognize that cotangent is the reciprocal of the tangent function. So, cot(θ) is equal to 1/tan(θ).
The tangent of an angle in the unit circle is equal to the ratio of the y-coordinate to the x-coordinate of the point where the terminal side of the angle intersects the circle. For an angle of -π/3, which is equivalent to rotating π/3 radians in the clockwise direction, we find ourselves in the fourth quadrant. In this quadrant, the x-coordinate is positive and the y-coordinate is negative. For the reference angle π/3, which is 60 degrees, the coordinates are (1/2, -√3/2). Therefore, tan(-π/3) will be the negative of the tangent of π/3, which is -(-√3/2)/(1/2) or √3.
The cotangent of -π/3 is therefore the reciprocal of √3, which is 1/√3. To rationalize the denominator, we multiply the numerator and denominator by √3, giving us √3/3. This is the exact value of cot(-π/3).