Finding the Coordinates of a Point on the Unit Circle Given an Angle

Given an angle \( \theta \) in radians, find the coordinates of the corresponding point on the unit circle.

Step 1: Recall the unit circle definition. The unit circle is a circle with a radius of 1 centered at the origin \((0, 0)\) in the coordinate plane.

Step 2: Use the cosine and sine functions to determine the coordinates: \(x = \cos(\theta)\), \(y = \sin(\theta)\).

Step 3: Compute the coordinates for a given angle \( \theta = \frac{5\pi}{6} \).

$x = \cos \left( \frac{5\pi}{6} \right) = -\frac{\sqrt{3}}{2}$

$y = \sin \left( \frac{5\pi}{6} \right) = \frac{1}{2}$

Therefore, the coordinates are \( \left( -\frac{\sqrt{3}}{2}, \frac{1}{2} \right) \).

Given an angle ( heta ) in radians, find the coordinates of the corresponding point on the unit circle.

Step 1: Recall the unit circle definition. The unit circle is a circle with a radius of 1 centered at the origin ((0, 0)) in the coordinate plane.

Step 2: Find the coordinates using (x = cos( heta)), (y = sin( heta)).

Step 3: For ( heta = frac{7pi}{4} ),

$x = cos left( frac{7pi}{4}
ight) = frac{sqrt{2}}{2}$

$y = sin left( frac{7pi}{4}
ight) = -frac{sqrt{2}}{2}$

Thus, the coordinates are ( left( frac{sqrt{2}}{2}, -frac{sqrt{2}}{2}
ight) ).

Given an angle ( heta ) in radians, find the coordinates of the corresponding point on the unit circle.

For ( heta = frac{11pi}{6} ),

$x = cos left( frac{11pi}{6}
ight) = frac{sqrt{3}}{2}$

$y = sin left( frac{11pi}{6}
ight) = -frac{1}{2}$

Therefore, the coordinates are ( left( frac{sqrt{3}}{2}, -frac{1}{2}
ight) ).

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