How do you simplify f(theta)=sec(theta/4)-cos(theta/4-pi/2) to trigonometric functions of a unit theta?

May 18, 2018

f(theta)=+-sqrt(2/(1+-sqrt((1+cos(theta))/2))) +-sqrt((1+-sqrt((1+cos(theta))/2))/2

Explanation:

In order to simplify what we have here, we have to use the Half-Angle identities and Sum and Difference formula.

First of all, let's simplify $\cos \left(\frac{\theta}{4} - \frac{\pi}{2}\right)$. Knowing that

$\cos \left(a - b\right) = \cos a \cos b + \sin a \sin b$

$\implies \cos \left(\frac{\theta}{4} - \frac{\pi}{2}\right) = \cos \left(\frac{\theta}{4}\right) \cos \left(\frac{\pi}{2}\right) + \sin \left(\frac{\theta}{4}\right) \cos \left(\frac{\pi}{2}\right)$

$\cos \left(\frac{\theta}{4} - \frac{\pi}{2}\right) = \cos \left(\frac{\theta}{4}\right) \cdot 0 + \sin \left(\frac{\theta}{4}\right) \cdot 1 = \sin \left(\frac{\theta}{4}\right)$

$\therefore \cos \left(\frac{\theta}{4} - \frac{\pi}{2}\right) = \sin \left(\frac{\theta}{4}\right)$

The Half-Angle identities state that

$\left\{\begin{matrix}\cos \left(\frac{x}{2}\right) = \pm \sqrt{\frac{1 + \cos x}{2}} \text{ [1]" \\ sin(x/2)=+-sqrt((1-cosx)/2)" [2]}\end{matrix}\right.$

The sign of the function we wish to have is given by the quadrant the angle $x$ is in.

Let us keep simplifying the second part of the function;

"[2]" => sin(theta/4)=+-sqrt((1-cos(theta"/"2))/2)

"[1]" => +-sqrt((1-cos(theta"/"2))/2)=+-sqrt((1+-sqrt((1+cos(theta))/2))/2

color(blue)( :. sin(theta/4)=+-sqrt((1+-sqrt((1+cos(theta))/2))/2

Now, let's pay our focus on $\sec \left(\theta \text{/} 4\right)$.

Knowing that $\sec x = \frac{1}{\cos} x$, we have

$\sec \left(\frac{\theta}{4}\right) = \frac{1}{\cos} \left(\frac{\theta}{4}\right)$

"[1]" => 1/cos(theta/4) = +-1/sqrt((1+cos(theta"/"2))/2)=+-sqrt(2/(1+cos(theta"/"2))

"[1]" => +-sqrt(2/(1+cos(theta"/"2))) = +-sqrt(2/(1+-sqrt((1+cos(theta))/2

color(blue)( :. sec(theta/4) = +-sqrt(2/(1+-sqrt((1+cos(theta))/2)

Finally, we can write $f \left(\theta\right)$ in terms of the unit $\theta$ as

color(blue)(f(theta) = +-sqrt(2/(1+-sqrt((1+cos(theta))/2))) +-sqrt((1+-sqrt((1+cos(theta))/2))/2

Note: Because we had $1 - \cos \left(\theta \text{/} 2\right)$ in the representation of $\sin \left(\theta \text{/} 4\right)$ and $1 + \cos \left(\theta \text{/} 2\right)$ in the representation of $\sec \left(\theta \text{/} 4\right)$, whenever we have + in the first expression, we will have - in the second one, such as:

$\sec \left(\frac{\theta}{4}\right) = \pm \sqrt{\frac{2}{1 \textcolor{red}{+} \sqrt{\frac{1 + \cos \left(\theta\right)}{2}}}}$

We must have

sin(theta/4) = +-sqrt((1color(red)-sqrt((1+cos(theta))/2))/2

And vice versa.