The terminal side of an angle is the angle at the standard position. Given that the point of intersection of the terminal side of θ and the unit circle is:
[tex](\frac{11}{23},y)[/tex]. The exact values of sine and cosine are: [tex]\sin(\theta) = \frac{2\sqrt{102}}{23}[/tex] and [tex]\cos(\theta) =\frac{11}{23}[/tex]
The given parameters are represented as:
[tex](\frac{11}{23},y)[/tex]
This means that:
[tex](\cos(\theta),\sin(\theta)) = (\frac{11}{23},y)[/tex]
Using the following trigonometry identity:
[tex]\cos^2(\theta) + \sin^2(\theta) = 1[/tex]
We have:
[tex](\frac{11}{23})^2+y^2=1[/tex]
Expand fraction
[tex]\frac{121}{529}+y^2=1[/tex]
Collect like terms
[tex]y^2=1 - \frac{121}{529}[/tex]
Take LCM
[tex]y^2=\frac{529 - 121}{529}[/tex]
[tex]y^2=\frac{408}{529}[/tex]
Take square roots
[tex]y=\frac{\sqrt{408}}{23}[/tex]
Substitute value for y in [tex](\cos(\theta),\sin(\theta)) = (\frac{11}{23},y)[/tex]
[tex](\cos(\theta),\sin(\theta)) = (\frac{11}{23},\frac{\sqrt{408}}{23})[/tex]
By comparison:
[tex]\cos(\theta) =\frac{11}{23}[/tex]
[tex]\sin(\theta) = \frac{\sqrt{408}}{23}[/tex]
Expand
[tex]\sin(\theta) = \frac{\sqrt{4*102}}{23}[/tex]
Split
[tex]\sin(\theta) = \frac{\sqrt{4}*\sqrt{102}}{23}[/tex]
Because the angle is in the first quadrant, we take only positive values
[tex]\sin(\theta) = \frac{2\sqrt{102}}{23}[/tex]
So, the exact values of sine and cosine are:
[tex]\sin(\theta) = \frac{2\sqrt{102}}{23}[/tex] and [tex]\cos(\theta) =\frac{11}{23}[/tex]
Read more about terminal angles at:
https://brainly.com/question/12891381
