Sec (23π/6)
sec(23π/6)
=sec(23π/6 ± 2kπ) k integer
=sec(23π/6 - 24π/6)
=sec(-π/6)
=1/cos(π/6) (cos(-x)=cos(x), sec=1/cos)
=(2/3)√3
To simplify the expression sec(23π/6), let's first determine the reference angle and the corresponding angle in the first quadrant.
The angle 23π/6 is larger than 2π (a full revolution), so we can subtract 2π from it to bring it into a range of one full revolution.
23π/6 - 2π = 11π/6
Now we have an angle of 11π/6, which is still greater than π (180 degrees). By subtracting π from it, we can find the equivalent angle within one revolution.
11π/6 - π = 5π/6
Now, we have an angle of 5π/6, which is in the second quadrant. To find the secant value, we can make use of the relationship between secant and cosine.
cos(5π/6) = 1 / sec(5π/6)
We know that cosine of 5π/6 is -√3/2, so we can rewrite the equation as follows:
-√3/2 = 1 / sec(5π/6)
To find sec(5π/6), we can take the reciprocal of -√3/2:
-2/√3 = sec(5π/6)
To simplify this further, we can rationalize the denominator by multiplying the numerator and denominator by √3:
(-2/√3) * (√3/√3) = -2√3 / 3
Therefore, sec(23π/6) equals -2√3 / 3.