# How do you evaluate sec(cos^-1(1/2)) without a calculator?

Oct 23, 2016

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

#### Explanation:

Another way, without calculating ${\cos}^{- 1} \left(\frac{1}{2}\right)$:

Consider a right triangle with an angle $\theta = {\cos}^{- 1} \left(\frac{1}{2}\right)$. Then $\cos \left(\theta\right) = \frac{1}{2}$, meaning the ratio of its adjacent side to the hypotenuse is $\frac{1}{2}$. Thus the ratio of the hypotenuse to its adjacent side, that is, $\sec \left(\theta\right)$, is $\frac{2}{1} = 2$.

Thus $\sec \left({\cos}^{- 1} \left(\frac{1}{2}\right)\right) = \sec \left(\theta\right) = 2$

Note that this same reasoning shows that in general, $\sec \left({\cos}^{- 1} \left(x\right)\right) = \frac{1}{x}$

Mar 12, 2017

$2$

#### Explanation:

Understand that color(blue)(cos^-1(1/2)=cos(theta)=(1/2)

As we know that, cosine function is the reciprocal of secant function,

color(brown)(sec(theta)=1/(cos(theta))

$\rightarrow \sec \left(\theta\right) = \frac{1}{\frac{1}{2}}$

$\Rightarrow \sec \left(\theta\right) = 2$

Hope this helps... :)