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
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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.