Answer:
Approximately [tex](-0.867)[/tex].
Step-by-step explanation:
The sine of this angle can be found in the following steps:
By the Pythagorean identity, for a particular angle [tex]\theta[/tex]:
[tex](\sin(\theta))^{2} + (\cos(\theta))^{2} = 1[/tex].
In this question, it is given that [tex]\cos(\theta) = (1/2)[/tex]. Rearrange the equation to find the absolute value of [tex]\sin(\theta)[/tex]:
[tex]\begin{aligned} |\sin(\theta)| &= \sqrt{1 - (\cos(\theta))^{2}} \\ &= \sqrt{1 - \left(\frac{1}{2}\right)^{2}} \\ &= \frac{\sqrt{3}}{2}\end{aligned}[/tex].
Since this angle is between [tex](3\pi / 2)[/tex] and [tex]2\, \pi[/tex], this angle is in the lower-right quadrant of the cartesian plane. Since that quadrant is below the horizontal [tex]x[/tex]-axis, [tex]y[/tex]-values in that quadrant would be negative. The sine value of the angle would also be negative.
The absolute value of a negative number is equal to the opposite of that number. Hence, the value of the sine of this angle would be the opposite of its absolute value:
[tex]\begin{aligned} \sin(\theta) &= -| \sin(\theta)| = -\frac{\sqrt{3}}{2} \approx -0.867\end{aligned}[/tex].
