What is the exact value of sec (7( pi))/6?

well, sec(theta-PI)=-secTheta

so that means sec(7PI/6)=-secPI/6= -1/cosPI/6= -3/sqrt3

the answer is -2/sqrt3

To find the exact value of sec(7π/6), we can use the unit circle and the definition of the secant function. The secant function is the reciprocal of the cosine function, so we need to find the cosine of 7π/6.

First, let's locate the angle 7π/6 on the unit circle. Starting from the positive x-axis, count clockwise (since the angle is negative) by 7/6 of a full revolution, which is equivalent to 210 degrees.

The angle 7π/6 is in the third quadrant of the unit circle, where the x-coordinate is negative. In this quadrant, the corresponding angle in the first quadrant is π/6, which has a known cosine value.

Cos(π/6) = √3/2

Since sec(x) is the reciprocal of cos(x), we can simply take the reciprocal of √3/2 to get the value of sec(7π/6).

sec(7π/6) = 1 / (√3/2) = 2 / √3

Now, to rationalize the denominator, we can multiply both the numerator and denominator by √3 to get rid of the square root:

sec(7π/6) = (2 / √3) * (√3 / √3) = 2√3 / 3

Therefore, the exact value of sec(7π/6) is 2√3 / 3.