- #1

sooyong94

- 173

- 2

## Homework Statement

A cubic equation is given as:

##x^{3} -(1+\cos \theta +\sin \theta)x^{2} +(\cos \theta \sin \theta +\cos \theta +\sin \theta)x-\sin \theta \cos \theta=0##

Show that x=1 is a root of the equation for all values of θ and deduce that x-1 is a factor to the above equation.

Hence, by factoring the cubic equation above, show that

##(x-1)[x^{2}-(\cos \theta +\sin \theta)x+\cos \theta \sin \theta]=0##

and the roots are given by

##1, \cos \theta, \sin \theta##

Write down the roots of the equation given that ##\theta=\frac{\pi}{3}##

Find all values of ##\theta## in the range ##0<=\theta<2\pi## such that two of the three roots are equal.

By considering ##\sin \theta -\cos \theta##, or otherwise, determine the greatest possible difference between the two roots, and find the values of ##\theta## for ##0<=\theta<2\pi## for which the two roots have the greatest difference.

## Homework Equations

Factor theorem, trigonometric equations

## The Attempt at a Solution

For the first part, I have plugged in x=1 and found that it is 0. Then I deduced that (x-1) is a factor.

I factored the cubic above and factored the quadratic, and the roots are 1, ##\cos \theta## and ##\sin \theta##. Then I plugged in ##\theta=\frac{\pi}{3}## and found out the solutions are

##1, \frac{1}{2}, \frac{sqrt{3}}{2}##. Is it correct?

Since the two roots are equal, therefore I set the following equations:

##\cos \theta =1##

##\sin \theta=1##

##\cos \theta=\sin \theta##

The values were found out to be 0, ##\frac{\pi}{4}##, ##\frac{\pi}{2}## and ##\frac{5\pi}{4}##. Hopefully I did not make any mistakes here... :P

For the last part, I need to consider ##\sin \theta-\cos \theta##. But how do I find the difference between the two roots?