To find the exact value of the trigonometric functions, specifically tangent, for the given angles (5Ï€/6, 7Ï€/6, and 11Ï€/6), we will use the unit circle. The unit circle represents the values of trigonometric functions for different angles.
First, let's consider the angle 5Ï€/6:
1. Draw the unit circle. The unit circle is a circle with a radius of 1 and is centered at the origin (0, 0) of the coordinate plane.
2. Measure the angle 5Ï€/6 on the unit circle in the counterclockwise direction from the positive x-axis. It will be in the second quadrant.
3. The point on the unit circle that corresponds to the angle 5π/6 is (-√3/2, 1/2).
Now, let's find the tangent of 5Ï€/6:
The tangent function is defined as the ratio of the sine of an angle to the cosine of that angle.
In this case, tan(5Ï€/6) = sin(5Ï€/6) / cos(5Ï€/6).
1. The sine of 5Ï€/6 is the y-coordinate of the point on the unit circle, which is 1/2.
2. The cosine of 5π/6 is the x-coordinate of the point on the unit circle, which is -√3/2.
Therefore, tan(5π/6) = (1/2) / (-√3/2).
To simplify this expression, we can multiply the numerator and denominator by 2/√3:
tan(5π/6) = (1/2) / (-√3/2) * (2/√3)/(2/√3)
= -√3 / 3.
Hence, the exact value of tan(5π/6) is -√3 / 3.
Similarly, we can find the exact values of tan(7Ï€/6) and tan(11Ï€/6).
For tan(7Ï€/6):
1. Measure the angle 7Ï€/6 on the unit circle. It will be in the third quadrant.
2. The point on the unit circle that corresponds to the angle 7π/6 is (√3/2, -1/2).
3. tan(7Ï€/6) = sin(7Ï€/6) / cos(7Ï€/6).
= (-1/2) / (√3/2).
= -√3 / 3.
For tan(11Ï€/6):
1. Measure the angle 11Ï€/6 on the unit circle. It will be in the fourth quadrant.
2. The point on the unit circle that corresponds to the angle 11π/6 is (-√3/2, -1/2).
3. tan(11Ï€/6) = sin(11Ï€/6) / cos(11Ï€/6).
= (-1/2) / (-√3/2).
= √3 / 3.
Hence, the exact values of tan(7π/6) and tan(11π/6) are also -√3/3 and √3/3, respectively.