# How do you get the exact value of sec^-1(-2)?

Sep 20, 2015

$\frac{2 \pi}{3} + 2 \pi n , \frac{4 \pi}{3} + 2 \pi n$

#### Explanation:

When working with inverse trig functions, it is better to reverse engineer slightly before you actually evaluate them. In your particular case, this would be rewriting as follows:

$\sec \left(x\right) = - 2$

Keep in mind that you could use any variable for x, I just chose x out of personal preference.

Now, because I've memorised the unit circle, I find it easier to work with sine, cosine and tangent functions. Therefore, I always want to try and get those functions. So, I will rewrite this as:

$\frac{1}{\cos} \left(x\right) = - 2$

Now, if I just go ahead and do some algebra, I get:

$\cos \left(x\right) = - \frac{1}{2}$

Look familiar? Now we could stop right here and use our unit circle, but since we're talking about inverse trig, I will take it forward just one more step:

$x = {\cos}^{-} 1 \left(- \frac{1}{2}\right)$

The final answer to this would be $\frac{2 \pi}{3} + 2 \pi n , \frac{4 \pi}{3} + 2 \pi n$

If you're unsure how we derived this final answer with the unit circle, or have trouble memorising it, I'd encourage you to watch my video .

Hope that helped :)