Given the condition that [tex]\(\cot \theta\)[/tex] is undefined and [tex]\(\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}\)[/tex], we can infer some key properties about [tex]\(\theta\)[/tex]. Here's a detailed step-by-step solution:
1. Understanding [tex]\(\cot \theta\)[/tex]:
- The cotangent function is defined as [tex]\(\cot(\theta) = \frac{\cos(\theta)}{\sin(\theta)}\)[/tex].
- If [tex]\(\cot \theta\)[/tex] is undefined, it means the denominator of the fraction must be zero, i.e., [tex]\(\sin(\theta) = 0\)[/tex]. However, this directly contradicts with the range [tex]\(\frac{\pi}{2} \leq \theta \leq \frac{3\pi}{2}\)[/tex] where sine is never zero within this interval but rather at the boundaries the angle [tex]\(\pi/2\)[/tex] and [tex]\(3\pi/2\)[/tex].
2. Analyzing [tex]\(\theta = \pi/2\)[/tex]:
- For [tex]\(\theta = \pi/2\)[/tex]:
- [tex]\(\sin(\pi/2) = 1\)[/tex]
- [tex]\(\cos(\pi/2) = 0\)[/tex]
- [tex]\(\tan(\pi/2)\)[/tex] is undefined because [tex]\(\tan(\pi/2) = \frac{\sin(\pi/2)}{\cos(\pi/2)} = \frac{1}{0}\)[/tex], which is a division by zero.
- The cosecant function [tex]\( \csc(\theta) = \frac{1}{\sin(\theta)} \)[/tex]. Therefore, [tex]\(\csc(\pi/2) = \frac{1}{1} = 1\)[/tex]
- The secant function [tex]\( \sec(\theta) = \frac{1}{\cos(\theta)} \)[/tex]. Hence, [tex]\(\sec(\pi/2) = \frac{1}{0}\)[/tex], which is undefined due to division by zero.
With this in mind, we fill in the values from the required trigonometric functions:
- [tex]\(\sin \theta = 1\)[/tex]
- [tex]\(\cos \theta = 6.123233995736766 \times 10^{-17}\)[/tex] (This value is extremely close to zero, and it arises from the limitations of numerical precision in computing [tex]\(\cos(\pi/2)\)[/tex])
- [tex]\(\tan \theta\)[/tex] is undefined
- [tex]\(\csc \theta = 1\)[/tex]
- [tex]\(\sec \theta\)[/tex] is undefined
These values are consistent with trigonometric function definitions at [tex]\(\theta = \pi/2\)[/tex]. The tiny non-zero value for [tex]\(\cos \theta\)[/tex] is due to computational approximation and can be considered practically zero.
