How do you simplify #f(theta)=-2csc(theta/4)+tan(theta/2)-3cos(theta/4)# to trigonometric functions of a unit #theta#?

Question

Anna Andrews · Accepted Answer

#f(theta)=-2csc(theta/4)+tan(theta/2)-3cos(theta/4)# #=frac{-2}{sin(theta/4)}+frac{sin(theta/2)}{cos(theta/2)}-3cos(theta/4)# #=frac{-2cos(theta/2)+sin(theta/2)sin(theta/4)-3cos(theta/4)sin(theta/4)cos(theta/2)}{sin(theta/4)cos(theta/2)}# Some identities to substitute in (half-angle and quarter-angle formulas):
#sin(a/2)=+-sqrt[frac{1-cosa}{2}]#
#cos(a/2)=+-sqrt[frac{1+cosa}{2}]#
#sin(theta/4)=sin((theta/2)/2)=+-sqrt[frac{1-+-sqrtfrac{1+costheta}{2}}{2}]#
#cos(theta/4)=cos((theta/2)/2)=+-sqrtfrac{1++-sqrtfrac{1-costheta}{2}}{2}#

Josiah Anderson · Answer

undefined To simplify $ f(\theta) = -2\csc\left(\frac{\theta}{4}\right) + \tan\left(\frac{\theta}{2}\right) - 3\cos\left(\frac{\theta}{4}\right) $ to trigonometric functions of a unit theta:

1. Recall the following trigonometric identities:
   - $ \csc(\theta) = \frac{1}{\sin(\theta)} $
   - $ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $
   - $ \cos(2\theta) = 1 - 2\sin^2(\theta) $

2. Substitute the identities into $ f(\theta) $ and simplify:
   $ f(\theta) = -2\left(\frac{1}{\sin\left(\frac{\theta}{4}\right)}\right) + \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} - 3\cos\left(\frac{\theta}{4}\right) $
   
   $ f(\theta) = -\frac{2}{\sin\left(\frac{\theta}{4}\right)} + \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} - 3\cos\left(\frac{\theta}{4}\right) $
   
   $ f(\theta) = -\frac{2}{\sin\left(\frac{\theta}{4}\right)} + \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} - 3\sqrt{1 - \sin^2\left(\frac{\theta}{4}\right)} $
   
3. Rationalize the denominator of the first term by multiplying numerator and denominator by $ \sin\left(\frac{\theta}{4}\right) $:
   $ f(\theta) = -\frac{2\sin\left(\frac{\theta}{4}\right)}{\sin^2\left(\frac{\theta}{4}\right)} + \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} - 3\sqrt{1 - \sin^2\left(\frac{\theta}{4}\right)} $
   
   $ f(\theta) = -\frac{2\sin\left(\frac{\theta}{4}\right)}{\sin^2\left(\frac{\theta}{4}\right)} + \frac{\sin\left(\frac{\theta}{2}\right)}{\cos\left(\frac{\theta}{2}\right)} - 3\sqrt{\cos^2\left(\frac{\theta}{4}\right)} $
   
4. Simplify further as needed, using trigonometric identities.