exact value for tan(7pi/6 radians)

That would be the same as tan (pi/6). (Tangent is positive inthe third quadrant, where 7pi/6 is located).

pi/6 is 1/3 of a right angle, or 30 degrees. You probably know that the tangent of that angle is 1/sqrt3

tan7pi/6

To find the exact value of tan(7π/6) radians, we can use the unit circle and trigonometric identities.

The angle 7π/6 radians lies in the third quadrant on the unit circle.

In the third quadrant, the reference angle (the acute angle between the terminal side of the angle and the x-axis) is π/6 radians.

Since the tangent function is negative in the third quadrant, we can find the exact value of tan(π/6) and then take its negative.

The reference angle of π/6 radians corresponds to a 30-degree angle which lies on the unit circle. In the unit circle, we can see that the point corresponding to the angle π/6 radians is (√3/2, -1/2).

Now, to calculate the exact value of tan(π/6), we use the formula:

tan(theta) = sin(theta) / cos(theta)

Using the coordinates from the unit circle, sin(π/6) = -1/2, and cos(π/6) = √3/2.

Therefore, tan(π/6) = (-1/2) / (√3/2) = -1 / √3 = -√3 / 3.

Finally, to find tan(7π/6), we take the negative of the value we found for tan(π/6), which gives us:

tan(7π/6) = - (-√3 / 3) = √3 / 3.

So, the exact value of tan(7π/6) radians is √3 / 3.