What is the cotangent of pi/2?
Understand the Problem
The question is asking for the value of the cotangent function at the angle pi/2 radians. Since cotangent is defined as the ratio of the cosine to the sine, we will need to evaluate this ratio at the specified angle.
Answer
undefined
Answer for screen readers
The value of cotangent at $\pi/2$ is undefined.
Steps to Solve
- Evaluate sine and cosine at $\pi/2$ radians
At $\pi/2$ radians, the values of sine and cosine functions are:
$$\sin(\pi/2) = 1$$ $$\cos(\pi/2) = 0$$
- Define cotangent function
The cotangent function is defined as the ratio of the cosine to the sine:
$$\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}$$
- Substitute the values into the cotangent function
Substitute $\pi/2$ into the cotangent function:
$$\cot(\pi/2) = \frac{\cos(\pi/2)}{\sin(\pi/2)} = \frac{0}{1}$$
- Simplify the expression
Simplify the fraction:
$$\cot(\pi/2) = 0$$
However, since division by zero is undefined, it's important to recognize that the cotangent function has no defined value at this angle.
The correct notation is:
$$\cot(\pi/2) = \text{undefined}$$
The value of cotangent at $\pi/2$ is undefined.
More Information
The cotangent function is undefined where the sine function is zero because cotangent is the ratio of cosine to sine. At $\pi/2$ radians, sine is 1, and cosine is 0, leading to an undefined expression.
Tips
A common mistake is to conclude the cotangent value is zero because cosine is zero. Always remember to check the denominator to avoid dividing by zero.
