The ordered pair of a unit circle is represented as:
[tex](x,y) \to (\sin(\theta),\cos(\theta))[/tex]
The point where the terminal side of [tex]\tehta[/tex][tex]\theta[/tex] intersect with the unit circle is: 240
The given parameter can be represented as:
[tex](x,y) \to (-\frac{\sqrt 3}{2},-\frac{1}{2})[/tex]
This means that:
[tex]\sin(\theta) = -\frac{\sqrt 3}{2}[/tex]
[tex]\cos(\theta) = -\frac{1}{2}[/tex]
Let the angle between the x-axis and the terminal side be [tex]\alpha[/tex]
So, we have:
[tex]\theta = 180 + \alpha[/tex]
Using [tex]\cos(\theta) = -\frac{1}{2}[/tex]
The equation becomes
[tex]\cos(180 + \alpha) = -\frac{1}{2}[/tex]
Apply cosine formula
[tex]\cos(180)*\cos(\alpha) - \sin(180)*\sin(\alpha) = -\frac{1}{2}[/tex]
[tex]-1*\cos(\alpha) - 0*\sin(\alpha) = -\frac{1}{2}[/tex]
[tex]-1*\cos(\alpha)= -\frac{1}{2}[/tex]
Divide both sides by -1
[tex]\cos(\alpha)= \frac{1}{2}[/tex]
Take arccos of both sides
[tex]\alpha= \cos^{-1}(\frac{1}{2})[/tex]
[tex]\alpha= \cos^{-1}(0.5)[/tex]
[tex]\alpha= 60[/tex]
Recall that:
[tex]\theta = 180 + \alpha[/tex]
[tex]\theta = 180 +60[/tex]
[tex]\theta = 240[/tex]
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