Find the value of $sec( heta)$ using the unit circle when $ heta = frac{2pi}{3}$, and verify the result using three different methods.

First, we find the coordinates of the point on the unit circle corresponding to $\theta = \frac{2\pi}{3}$.

The coordinates are $\left( -\frac{1}{2}, \frac{\sqrt{3}}{2} \right)$.

Since $\sec(\theta) = \frac{1}{\cos(\theta)}$, we have:

$\sec\left(\frac{2\pi}{3}\right) = \frac{1}{\cos\left(\frac{2\pi}{3}\right)} = \frac{1}{-\frac{1}{2}} = -2.$

Verification using the Pythagorean identity:

$\sec^2(\theta) = 1 + \tan^2(\theta)$

$\tan\left(\frac{2\pi}{3}\right) = -\sqrt{3}$

$\sec^2\left(\frac{2\pi}{3}\right) = 1 + 3 = 4$

$\sec\left(\frac{2\pi}{3}\right) = \pm 2 = -2.$

Let’s verify sec(θ) using the reference angle method:

The reference angle for $ heta = frac{2pi}{3}$ is $frac{pi}{3}$, and we know $cosleft(frac{pi}{3}
ight) = frac{1}{2}$.

Since $ heta$ is in the second quadrant, $cosleft(frac{2pi}{3}
ight)$ is negative:

$cosleft(frac{2pi}{3}
ight) = -cosleft(frac{pi}{3}
ight) = -frac{1}{2}.$

Therefore,

$secleft(frac{2pi}{3}
ight) = frac{1}{-frac{1}{2}} = -2.$

Using the symmetry of the unit circle, we note that $cosleft(pi – x
ight) = -cos(x)$:

For $ heta = frac{2pi}{3}$,

$cosleft(frac{2pi}{3}
ight) = -cosleft(frac{pi}{3}
ight) = -frac{1}{2}.$

So,

$secleft(frac{2pi}{3}
ight) = frac{1}{cosleft(frac{2pi}{3}
ight)} = frac{1}{-frac{1}{2}} = -2.$

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