Find solutions of an equation within an interval

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May 15th 2011, 11:58 AM
root

Find solutions of an equation within an interval

My class has moved on to a new section of trigonometry that I am stuck on.

I am suppose to find all solutions of the equation in the interval [0,2π]

An example problem that is in my text book has a problem with its corresponding answer below, but I don't really follow.

Example: cos³ x = cos x
Answer: 0, π/2, π, 3π/2

It would be really helpful if you can explain the steps that you take in order to get the final answer.

Also π = pi.

Thanks in advance
May 15th 2011, 12:14 PM
TheEmptySet

Quote:

Originally Posted by root

My class has moved on to a new section of trigonometry that I am stuck on.

I am suppose to find all solutions of the equation in the interval [0,2π]

An example problem that is in my text book has a problem with its corresponding answer below, but I don't really follow.

Example: cos³ x = cos x
Answer: 0, π/2, π, 3π/2

It would be really helpful if you can explain the steps that you take in order to get the final answer.

Also π = pi.

Thanks in advance

If you subtract cosine from both sides you get

$\cos^3(x)-\cos(x)=0$

Now factor the left hand side completely to get

$\cos(x)[\cos^2(x)-1]=0 \iff \cos(x)[\cos(x)-1][\cos(x)+1]=0$

The last step comes from the difference of squares.

Now use the zero factor principle to set each factor equal to zero and solve.

$\cos(x)=0 \quad \cos(x)-1=0\quad \cos(x)+1=0$
May 15th 2011, 12:35 PM
root

Thank you

Thank you for replying so quickly. Your explanation was a lot more helpful than my textbook's. I get it now.