In trigonometry, what is the exact value of cot(5π/3) expressed in simplest radical form?

Introduction / Context:
This problem tests your understanding of trigonometric ratios for standard angles on the unit circle, particularly in radians. The function cotangent, or cot, is the reciprocal of tangent and can be expressed in terms of sine and cosine. Here, you are asked to find the exact value of cot(5π/3), a common angle in trigonometry. Knowing coordinates of standard angles and the signs of trigonometric functions in each quadrant is essential for this type of question.

Given Data / Assumptions:

• The angle is 5π/3 radians.
• cot theta is defined as cos theta divided by sin theta.
• We use the standard unit circle values for sine and cosine of key angles.
• We seek an exact value, not a decimal approximation.

Concept / Approach:
First, interpret 5π/3 in degrees or as a position on the unit circle. 5π/3 radians corresponds to 300 degrees, which lies in the fourth quadrant. In this quadrant, cosine is positive and sine is negative. The reference angle is 60 degrees or π/3, whose sine and cosine are known. Using sin 300 degrees and cos 300 degrees, we can compute cot 300 degrees as cos divided by sin and then simplify the result to get a simple radical form.

Step-by-Step Solution:
Step 1: Convert 5π/3 to degrees if desired: 5π/3 radians equals 300 degrees. Step 2: Recognise that 300 degrees has a reference angle of 60 degrees, so we use values for sine and cosine of 60 degrees with appropriate signs. Step 3: For 300 degrees, cos 300 degrees = 1/2 and sin 300 degrees = -sqrt(3)/2. Step 4: Compute cot 300 degrees using cot theta = cos theta / sin theta: cot 300 degrees = (1/2) / (-sqrt(3)/2). Step 5: Simplify the fraction: (1/2) divided by (-sqrt(3)/2) equals (1/2) * (2 / -sqrt(3)) = -1 / sqrt(3).

Verification / Alternative check:
As a check, recall that tan 300 degrees is sin 300 degrees divided by cos 300 degrees, which equals (-sqrt(3)/2) / (1/2) = -sqrt(3). Since cotangent is the reciprocal of tangent, cot 300 degrees should be 1 / tan 300 degrees = 1 / (-sqrt(3)) = -1 / sqrt(3). This matches the previous calculation and confirms the correctness of the value.

Why Other Options Are Wrong:

• Option a (1/sqrt(3)) has the correct magnitude but the wrong sign; it describes cot 60 degrees, not cot 300 degrees.
• Option c (sqrt(3)) and option d (-sqrt(3)) are values associated with tangent rather than cotangent here.
• Option e (0) would correspond to an angle where sin equals infinity or cos equals zero in a way that makes tan infinite, not to cot 300 degrees.

Common Pitfalls:
A common error is to forget which trigonometric functions are positive in each quadrant and to assign the wrong sign to sine or cosine. Another mistake is to confuse tangent and cotangent or to take the reciprocal incorrectly. Some students also misidentify the reference angle. To avoid these issues, always determine the quadrant, write down the signs of sine and cosine there, use exact values for standard angles, and then compute the ratio carefully.

Final Answer:
The exact value of cot(5π/3) is -1/sqrt(3).