# Thread: Algebraic Expressions in Trig.

1. ## Algebraic Expressions in Trig.

Find an algebraic expression equivalent to tan ( arc cos X )

Evaluate:
arc cos ( cos pi/3)
arc cos ( cos -pi/7)

2. Originally Posted by Godfather
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

$\displaystyle tan(\theta)=\frac{opp}{adj}=\frac{\sqrt{1-x^2}}{x}$

3. 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.