π/3 (1/2, √3/2)
π/3 (1/2, √3/2) is located on the unit circle at an angle of π/3 radians (60 degrees) from the positive x-axis in the counterclockwise direction
The given information π/3 (1/2, √3/2) represents a point on the unit circle with angle π/3 radians or 60 degrees and coordinates (1/2, √3/2) on the circle.
To understand this, we should know what the unit circle is. The unit circle is a circle with a radius of 1 unit, centered at the origin of a coordinate plane. It is used to define trigonometric functions sine and cosine for any angle measure, including radians.
Now let’s break this down and see what we know from the given information:
Angle measure: The angle measure is π/3 radians or 60 degrees. This means that the point is located on the unit circle at an angle of 60 degrees from the positive x-axis (or 1st quadrant) in the counterclockwise direction.
Coordinates: The coordinates of the point are (1/2, √3/2). These coordinates are the x and y values of the point on the unit circle. We can use the Pythagorean theorem to find that the distance from the origin to the point is 1. We can also use the coordinates to find the values of sine and cosine functions of the angle.
Cosine value: The cosine function of the angle is given by the x-coordinate of the point, which is 1/2. Therefore, cos(π/3) = 1/2.
Sine value: The sine function of the angle is given by the y-coordinate of the point, which is √3/2. Therefore, sin(π/3) = √3/2.
In summary, the point π/3 (1/2, √3/2) is located on the unit circle at an angle of π/3 radians (60 degrees) from the positive x-axis in the counterclockwise direction. The cosine and sine values of this angle are 1/2 and √3/2, respectively.
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