To find the coordinates (x, y) on the unit circle that correspond to the real number t = 2π/3, follow these steps:
- Understanding the Unit Circle: The unit circle is a circle with a radius of 1 centered at the origin (0, 0) in the Cartesian coordinate system. Points on the unit circle can be represented in polar coordinates, where the angle t (in radians) determines a point on the circle.
- Calculating the Coordinates: The x and y coordinates on the unit circle can be expressed using the cosine and sine functions, respectively:
- x = cos(t)
- y = sin(t)
- Substituting the Value of t: Since we have t = 2π/3, we will substitute this value into the cosine and sine functions:
- x = cos(2π/3)
- y = sin(2π/3)
- Finding the Cosine and Sine: The angle 2π/3 is located in the second quadrant of the unit circle, where the x-values are negative, and the y-values are positive:
- Using the reference angle of π/3, we find:
- cos(2π/3) = -cos(π/3) = -1/2
- sin(2π/3) = sin(π/3) = √3/2
- Final Coordinates: Therefore, the coordinates (x, y) on the unit circle corresponding to the real number t = 2π/3 are:
- (x, y) = (-1/2, √3/2)
In summary, the point on the unit circle that corresponds to the angle 2π/3 is (-1/2, √3/2).