Find the exact value of thet following trigonometric functions:

tan(5pi/6)
tan(7pi/6)
tan(11pi/6)

You should be able to make a quick sketch of the following right-angled triangles

30° - 60° - 90° ---> π/6rad - π/3rad - π/2 ---> 1 -- √3 -- 2
and
45° - 45° - 90° --> π/4 - π/4 - π/2 ---> 1 - 1 - √2

You should also know the CAST rule, that is, where are each of the basic trig functions + or -

I will do the second one for you
tan (7π/6)
If it helps you, let's do in degrees.
7π/6 = 7(π/6) = 7(30°) = 210°
Now 210° is in quad III and 30 from the x-axis
we know from the above triangles that
tan 30° = 1/√3
so in the third quadrant, by CAST, the tangent is positive,
thus tan 7π/6 = 1/√3

( you can of course check all your answers with a calculator)

do the others the same way

To find the exact value of the trigonometric functions, specifically tangent, for the given angles (5π/6, 7π/6, and 11π/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 5π/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 5π/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 5π/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(5π/6) = sin(5π/6) / cos(5π/6).

1. The sine of 5π/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(7π/6) and tan(11π/6).

For tan(7π/6):
1. Measure the angle 7π/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(7π/6) = sin(7π/6) / cos(7π/6).
= (-1/2) / (√3/2).
= -√3 / 3.

For tan(11π/6):
1. Measure the angle 11π/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(11π/6) = sin(11π/6) / cos(11π/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.