correction: use the unit circle to fin the ecaxtvalue for all trig functions theta=2pi/3
To find the exact values of trigonometric functions, specifically for theta = 2pi/3, we will use the unit circle.
1. Draw a coordinate plane with the origin at the center (0,0) and a circle with a radius of 1 unit.
2. Divide the circle into four equal parts (quadrants) by drawing two perpendicular lines passing through the origin. Label the positive x-axis as "1", the positive y-axis as "2", the negative x-axis as "3", and the negative y-axis as "4".
3. Now, locate an angle of 2pi/3 on the unit circle. Remember that the given angle is measured in radians.
To find this angle, start from the positive x-axis in the counterclockwise direction. Rotate a quarter of a circle, which is pi/2 radians. Then, rotate an additional pi/3 radians, which is equivalent to 2pi/6. So, the angle 2pi/3 can be found by rotating a total of 5pi/6 radians.
4. The point where the terminal side of the angle intersects the unit circle is the point (x,y) on the circle.
To find the coordinates (x,y), we need to determine the trigonometric values for theta = 2pi/3.
The x-coordinate (x-value) corresponds to the cosine function, so we want to find the cos(2pi/3).
The y-coordinate (y-value) corresponds to the sine function, so we want to find the sin(2pi/3).
The tangent function is sin/cos, so we also want to find the tan(2pi/3).
5. To evaluate these trigonometric functions, we will use the properties specific to the angle 2pi/3 on the unit circle.
- The x-coordinate of the point is -1/2.
Therefore, cos(2pi/3) = -1/2.
- The y-coordinate of the point is sqrt(3)/2.
Therefore, sin(2pi/3) = sqrt(3)/2.
- The tangent of theta can be found by dividing sin(theta) by cos(theta).
Therefore, tan(2pi/3) = (sqrt(3)/2) / (-1/2) = -sqrt(3).
Thus, the exact values for the trigonometric functions when theta = 2pi/3 are:
cos(2pi/3) = -1/2
sin(2pi/3) = sqrt(3)/2
tan(2pi/3) = -sqrt(3)