The coordinates of the point on the unit circle at angle 225° are [tex]\bold{(-\frac{\sqrt{2} }{2}, -\frac{\sqrt{2} }{2})}[/tex]
What are the coordinates of the point on the circle?
The coordinates of the points of the circle x² + y² = r² are [tex](r~cos\theta, r~sin\theta)[/tex]
What is unit circle?
"It is a circle with radius 1."
For given example,
We have been given an angle of 225°
⇒ [tex]\theta[/tex] = 225°
We need to find the coordinates of the point on the unit circle at an angle 225°.
Here, r = 1
So, the first coordinate would be,
[tex]r~cos\theta=(1)~cos(225^{\circ})\\\\\Rightarrow r~cos(225^{\circ})=cos(270^{\circ} - 45^{\circ})\\\\\Rightarrow r~cos(225^{\circ})t=cos(270^{\circ}) cos(45^{\circ})+sin(270^{\circ}) sin(45^{\circ})\\\\\Rightarrow r~cos(225^{\circ})=0+(-1)(\frac{1}{\sqrt{2} } )\\\\\Rightarrow r~cos(225^{\circ})=\bold{\frac{-1}{\sqrt{2} }}\\\\\Rightarrow r~cos(225^{\circ})=\bold{-\frac{\sqrt{2} }{2} }[/tex]
Now we find the second coordinate of the point.
[tex]r~sin\theta=(1)~sin(225^{\circ})\\\\\Rightarrow r~sin(225^{\circ})=sin(270^{\circ} - 45^{\circ})\\\\\Rightarrow r~sin(225^{\circ})=sin(270^{\circ}) cos(45^{\circ})-cos(270^{\circ}) sin(45^{\circ})\\\\\Rightarrow r~sin(225^{\circ})=(-1)(\frac{1}{\sqrt{2} } )-0\\\\\Rightarrow r~sin(225^{\circ})=\bold{\frac{-1}{\sqrt{2} }}\\\\\Rightarrow r~sin(225^{\circ})=\bold{-\frac{\sqrt{2} }{2} }[/tex]
Therefore, the coordinates of the point on the unit circle are [tex](-\frac{\sqrt{2} }{2}, -\frac{\sqrt{2} }{2})[/tex]
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