To find a simplified expression for [tex]\(\sin\left(\cos^{-1}\left(\frac{t}{2}\right)\right)\)[/tex], we can leverage trigonometric identities and the properties of inverse trigonometric functions.
Let's start by setting [tex]\(\theta = \cos^{-1}\left(\frac{t}{2}\right)\)[/tex]. This means that:
[tex]\[ \cos(\theta) = \frac{t}{2} \][/tex]
Our goal is to find [tex]\(\sin(\theta)\)[/tex]. We can use the Pythagorean identity, which states:
[tex]\[ \sin^2(\theta) + \cos^2(\theta) = 1 \][/tex]
Substitute [tex]\(\cos(\theta) = \frac{t}{2}\)[/tex] into this identity:
[tex]\[ \sin^2(\theta) + \left(\frac{t}{2}\right)^2 = 1 \][/tex]
Simplify the expression inside the parenthesis:
[tex]\[ \sin^2(\theta) + \frac{t^2}{4} = 1 \][/tex]
To isolate [tex]\(\sin^2(\theta)\)[/tex], subtract [tex]\(\frac{t^2}{4}\)[/tex] from both sides:
[tex]\[ \sin^2(\theta) = 1 - \frac{t^2}{4} \][/tex]
Next, we need to take the square root of both sides to find [tex]\(\sin(\theta)\)[/tex]. Remember to consider the positive root because [tex]\(\theta = \cos^{-1}\left(\frac{t}{2}\right)\)[/tex] lies in the range [tex]\([0, \pi]\)[/tex] where the sine function is non-negative.
[tex]\[ \sin(\theta) = \sqrt{1 - \frac{t^2}{4}} \][/tex]
Alternatively, we can express the result with a common denominator inside the square root:
[tex]\[ \sin(\theta) = \sqrt{\frac{4 - t^2}{4}} \][/tex]
Which simplifies to:
[tex]\[ \sin(\theta) = \frac{\sqrt{4 - t^2}}{2} \][/tex]
Therefore, the simplified expression for [tex]\(\sin\left(\cos^{-1}\left(\frac{t}{2}\right)\right)\)[/tex] is:
[tex]\[ \sin\left(\cos^{-1}\left(\frac{t}{2}\right)\right) = \frac{\sqrt{4 - t^2}}{2} \][/tex]
This is the simplified expression you are looking for.
