Find an algebraic expression equivalent to tan ( arc cos X )
Evaluate:
arc cos ( cos pi/3)
arc cos ( cos -pi/7)
Draw a reference triangle using the defintion of the cosine function...
1 will be the hypotenuse and x will be the adjacent side....
solve for the opposite side using the pythagorean theorem.
$\displaystyle x^2+(opp)^2=1^2$
$\displaystyle opp=\sqrt{1-x^2}$
Using the definition of tangent gives
$\displaystyle tan(\theta)=\frac{opp}{adj}=\frac{\sqrt{1-x^2}}{x}$
For the second part....
We need to know the domain of the arc cosine function....
The domain is 0 to Pi.
also note that the cosine function is even ie
$\displaystyle cos(-\theta)=cos(\theta)$
This gives us
$\displaystyle cos\left(\frac{-\pi}{7}\right)=cos\left(\frac{\pi}{7}\right)$
so
$\displaystyle cos^{-1}(cos\left(\frac{\pi}{3}\right))=\frac{\pi}{3}$
and
$\displaystyle cos^{-1}(cos\left(\frac{-\pi}{7}\right))=cos^{-1}(cos\left(\frac{\pi}{7}\right))=\frac{\pi}{7}$
since the angle is now in the domain of the arc sine function.