To find [tex]\(\csc \left(\theta - \frac{\pi}{2}\right)\)[/tex] given that [tex]\(\cos \theta = 0.5\)[/tex], let's go through the problem step by step.
1. Determine [tex]\(\sin \theta\)[/tex]:
Since [tex]\(\cos^2 \theta + \sin^2 \theta = 1\)[/tex], we can solve for [tex]\(\sin \theta\)[/tex]:
[tex]\[
\sin \theta = \sqrt{1 - \cos^2 \theta}
\][/tex]
Given [tex]\(\cos \theta = 0.5\)[/tex], we substitute:
[tex]\[
\sin \theta = \sqrt{1 - (0.5)^2} = \sqrt{1 - 0.25} = \sqrt{0.75} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2} \approx 0.866
\][/tex]
2. Recall the sine and cosecant transformations:
The sine of the angle [tex]\(\left(\theta - \frac{\pi}{2}\right)\)[/tex] can be expressed using the co-function identity:
[tex]\[
\sin \left(\theta - \frac{\pi}{2}\right) = -\cos \theta
\][/tex]
Given [tex]\(\cos \theta = 0.5\)[/tex], we have:
[tex]\[
\sin \left(\theta - \frac{\pi}{2}\right) = -0.5
\][/tex]
3. Calculate the cosecant of the angle:
The cosecant is the reciprocal of the sine function:
[tex]\[
\csc \left(\theta - \frac{\pi}{2}\right) = \frac{1}{\sin \left(\theta - \frac{\pi}{2}\right)}
\][/tex]
Substitute [tex]\(\sin \left(\theta - \frac{\pi}{2}\right) = -0.5\)[/tex]:
[tex]\[
\csc \left(\theta - \frac{\pi}{2}\right) = \frac{1}{-0.5} = -2
\][/tex]
Therefore, [tex]\(\csc \left(\theta - \frac{\pi}{2}\right)\)[/tex] is [tex]\(-2\)[/tex].
