Find the exact value of each expression for the given value of theta. Sec^2(2theta) if theta = pi/6
To find the value of sec^2(2theta) when theta = pi/6, we first need to determine the value of 2theta when theta = pi/6.
Since 2theta = 2 * (pi/6) = (2pi/6) = pi/3, we can rewrite the expression as sec^2(pi/3).
To evaluate sec^2(pi/3), we need to find the value of sec(pi/3). Recall that sec(theta) is equal to the reciprocal of cos(theta), so sec(pi/3) = 1/cos(pi/3).
Using the unit circle or trigonometric identities, we know that cos(pi/3) = 1/2, so sec(pi/3) = 1/(1/2) = 2.
Finally, substituting this value into sec^2(pi/3), we get sec^2(pi/3) = (2)^2 = 4.
Therefore, sec^2(2theta) = 4 when theta = pi/6.