To find [tex]\(\sin(\theta)\)[/tex] given that [tex]\(\cos(\theta) = \frac{2}{7}\)[/tex] and the terminal side of [tex]\(\theta\)[/tex] lies in quadrant IV, you can use the Pythagorean identity. The Pythagorean identity states:
[tex]\[
\sin^2(\theta) + \cos^2(\theta) = 1
\][/tex]
Given [tex]\(\cos(\theta) = \frac{2}{7}\)[/tex], plug it into the identity:
[tex]\[
\sin^2(\theta) + \left(\frac{2}{7}\right)^2 = 1
\][/tex]
Calculate [tex]\(\left(\frac{2}{7}\right)^2\)[/tex]:
[tex]\[
\left(\frac{2}{7}\right)^2 = \frac{4}{49}
\][/tex]
Now, substitute it back into the identity:
[tex]\[
\sin^2(\theta) + \frac{4}{49} = 1
\][/tex]
To isolate [tex]\(\sin^2(\theta)\)[/tex], subtract [tex]\(\frac{4}{49}\)[/tex] from both sides:
[tex]\[
\sin^2(\theta) = 1 - \frac{4}{49}
\][/tex]
Rewrite 1 as a fraction with the same denominator:
[tex]\[
1 = \frac{49}{49}
\][/tex]
So,
[tex]\[
\sin^2(\theta) = \frac{49}{49} - \frac{4}{49} = \frac{45}{49}
\][/tex]
Next, take the square root of both sides to solve for [tex]\(\sin(\theta)\)[/tex]:
[tex]\[
\sin(\theta) = \pm \sqrt{\frac{45}{49}}
\][/tex]
Simplify the square root:
[tex]\[
\sin(\theta) = \pm \frac{\sqrt{45}}{7} = \pm \frac{3\sqrt{5}}{7}
\][/tex]
Since the terminal side of [tex]\(\theta\)[/tex] is in quadrant IV, and sine is negative in this quadrant, we select the negative value:
[tex]\[
\sin(\theta) = -\frac{3\sqrt{5}}{7}
\][/tex]
Therefore, the exact, fully simplified and rationalized answer is:
[tex]\[
\sin(\theta) = -\frac{3\sqrt{5}}{7}
\][/tex]
