Question

Determine the values of \theta if sec\;\theta=-\frac{2}{\sqrt{3}}.

Answer 1

Answer:

See below.

Step-by-step explanation:

So, we have:

$\sec(\theta)=-2/\sqrt{3}$

Recall that secant is simply the reciprocal of cosine. So we can:

$\cos(\theta)=(\sec(\theta))^{-1}=(-2/\sqrt{3})^{-1}\\\cos(\theta)=-\sqrt{3}/2$

Now, recall the unit circle. Since cosine is negative, it must be in Quadrants II and/or III. The numerator is the square root of 3. The denominator is 2. The whole thing is negative. Therefore, this means that 150 or 5π/6 is a candidate. Therefore, due to reference angles, 180+30=210 or 7π/6 is also a candidate.

Therefore, the possible values for theta is

5π/6 +2nπ

and

7π/6 + 2nπ