Evaluating trigonometric functions Without using a calculator, evaluate the following expressions or state that the quantity is undefined.


cot (-17π/3)

Verified step by step guidance

1. <Step 1: Understand the cotangent function. The cotangent function, \( \cot(\theta) \), is defined as \( \frac{\cos(\theta)}{\sin(\theta)} \).>
2. <Step 2: Simplify the angle. The angle \( -\frac{17\pi}{3} \) is not within the standard range \( [0, 2\pi) \). To simplify, add \( 2\pi \) repeatedly until the angle is within this range.>
3. <Step 3: Calculate the equivalent positive angle. Since \( 2\pi = \frac{6\pi}{3} \), add \( \frac{6\pi}{3} \) to \( -\frac{17\pi}{3} \) multiple times until the angle is positive and within \( [0, 2\pi) \).>
4. <Step 4: Determine the reference angle. Once the angle is simplified, find the reference angle, which is the acute angle the terminal side makes with the x-axis.>
5. <Step 5: Evaluate \( \cot(\theta) \) using the reference angle. Use known values of sine and cosine for the reference angle to find \( \cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)} \).>

Key Concepts

Cotangent Function

The cotangent function, denoted as cot(x), is the reciprocal of the tangent function. It is defined as cot(x) = cos(x)/sin(x). Understanding cotangent is essential for evaluating expressions involving this function, especially in trigonometric identities and transformations.

Periodic Properties of Trigonometric Functions

Trigonometric functions are periodic, meaning they repeat their values in regular intervals. For example, the cotangent function has a period of π, which means cot(x) = cot(x + nπ) for any integer n. This property allows us to simplify angles that are outside the standard range of 0 to 2π.

Angle Reduction

Angle reduction involves converting a given angle into an equivalent angle within a standard range, typically between 0 and 2π. For cot(-17π/3), we can add multiples of 2π to find a coterminal angle that is easier to evaluate, facilitating the calculation of the cotangent value.