Find the coordinates of a point on the unit circle corresponding to a given angle
Answer 1
Emma Johnson
Given an angle of \( \theta = 45^{\circ} \). To find the coordinates of the point on the unit circle:
The coordinates of any point on the unit circle can be found using the formulas:
\[ x = \cos(\theta) \]
\[ y = \sin(\theta) \]
Using \( \theta = 45^{\circ} \):
\[ x = \cos(45^{\circ}) = \cos \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \]
\[ y = \sin(45^{\circ}) = \sin \left( \frac{\pi}{4} \right) = \frac{\sqrt{2}}{2} \]
The coordinates are \( \left( \frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} \right) \).
Answer 2
James Taylor
Given an angle of ( heta = 60^{circ} ). To determine the coordinates of the point on the unit circle:
Recall the unit circle formulas:
[ x = cos( heta) ]
[ y = sin( heta) ]
Using ( heta = 60^{circ} ):
[ x = cos(60^{circ}) = cos left( frac{pi}{3}
ight) = frac{1}{2} ]
[ y = sin(60^{circ}) = sin left( frac{pi}{3}
ight) = frac{sqrt{3}}{2} ]
The coordinates are ( left( frac{1}{2}, frac{sqrt{3}}{2}
ight) ).
Answer 3
Emily Hall
Given an angle of ( heta = 30^{circ} ), find the coordinates on the unit circle:
Use the formulas:
[ x = cos( heta) ]
[ y = sin( heta) ]
Using ( heta = 30^{circ} ):
[ x = cos(30^{circ}) = frac{sqrt{3}}{2} ]
[ y = sin(30^{circ}) = frac{1}{2} ]
The coordinates are ( left( frac{sqrt{3}}{2}, frac{1}{2}
ight) ).
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