Cosecant of pi/2
Understand the Problem
The question is asking for the value of the cosecant function evaluated at the angle pi/2. Cosecant is the reciprocal of sine, so we need to determine the sine of pi/2 and then find its reciprocal.
Answer
The value of \( \csc\left(\frac{\pi}{2}\right) \) is 1.
Answer for screen readers
The value of ( \csc\left(\frac{\pi}{2}\right) ) is 1.
Steps to Solve
- Evaluate the sine of the angle
We need to find the sine of the angle ( \frac{\pi}{2} ).
The sine function at ( \frac{\pi}{2} ) is equal to 1: $$ \sin\left(\frac{\pi}{2}\right) = 1 $$
- Find the reciprocal for cosecant
Cosecant is the reciprocal of sine, so we find the cosecant by taking the reciprocal of the sine value calculated above: $$ \csc\left(\frac{\pi}{2}\right) = \frac{1}{\sin\left(\frac{\pi}{2}\right)} $$
- Calculate the cosecant value
Since ( \sin\left(\frac{\pi}{2}\right) = 1 ), we can compute: $$ \csc\left(\frac{\pi}{2}\right) = \frac{1}{1} = 1 $$
The value of ( \csc\left(\frac{\pi}{2}\right) ) is 1.
More Information
The cosecant function is one of the six fundamental trigonometric functions and is particularly important in various applications such as physics and engineering. The cosecant is undefined for angles where sine is zero, which does not include ( \frac{\pi}{2} ).
Tips
- Confusing sine and cosecant values. Remember that cosecant is the reciprocal of sine.
- Forgetting the value of ( \sin\left(\frac{\pi}{2}\right) ); this is crucial for correctly calculating cosecant.