Question 210322
I will rewrite this replacing theta with x to make it more clear:
{{{cos 3*x =(4 cos^3x)-(3cosx)}}}
I am going to work on the cos(3x) term to see if it can be made equal to the right side. The first step is kinda of unintuitive, yet it is the key to the problem. I am going to rewite cos(3x) as follows:
{{{cos3x=cos3x+cosx-cosx}}}
Notice this did not change the value of the term, since I added and subtracted the same thing (cos(x)). I did this because I want to apply the sum property, which gives me:
{{{cos3x=2cos2xcosx-cosx}}}
Now let's go double angle:
{{{cos3x=2(2cos^2(2x)-1)cosx-cosx}}}
Simplify:
{{{cos3x=4cos^2(x)-2)cosx-cosx}}}
{{{cos3x=4cos^3(x)-2cosx-cosx}}}
{{{highlight(cos3x=4cos^3(x)-3cosx)}}}
So there you have it... you may be discouraged by that first step and say "I would never think of that." Neither would I when I first started doing these, but if you do enough practice problems, you will see how these kind of manipulations can help you.
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Good Luck,
tutor_paul@yahoo.com