exam revise question i dont understand

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ChrisieB

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Oct 29, 2010
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hey guys!! i need help badly!!
i 'have exams coming up and the teacher has given us these revision questions but i dont understand them, i've completed the whole chapter except these ones are at the very end and are the very hardest
can someone help?

21. let S=1+2cosx+2cos2x+2cos3x

a) show that S (sin/2x) = sin(7x/2)

b) hence show that if x = 2(pie)/7, then 1+2cosx+2cos2x+2cos3x=0

c) by writing S in terms of cosx, prove that cos(2(pie)/7) is a solution to the plynomial equation 8x^3+4x^2-4x-1=0

22.
A) solve sin3x = cos2x, for 0<x<360

a) using solutions of cosa = cosb

B) use compound angle formulae in the above equation to find the exact values of sin18 and sin54
(Hint: 4u^3 - 2u^2-3u+1 = (u-1)(4u^2+2u-1)

C) use results from B to find the exact values of sin, cos and tan of 18 , 36, 54 and 72 degrees

thanks so much to ANYONE who can help me!!!
 
Please share your work with us, indicating exactly where you are stuck - so that we may know where to begin to help you.
 
i dont understand where to begin each question from. i dont understand how to break 1+2cosx+2cos2x+2cos3x and i dont understand how to solve sin3x=cos2x
 
Hello, ChrisieB!

\(\displaystyle \text{22. (A) Solve }\,\sin 3x \:=\: \cos2x,\,\text{ for }0^o < x < 360^o,\,\text{ using solutions of }\cos a = \cos b\) .??
Click to expand...

\(\displaystyle \text{If }\sin3x \,=\,\cos2x,\,\text{ the two angles are complementary.}\)

\(\displaystyle \text{Take a look . . .}\)

Code: 
                  *
                * *
              * 2x*
         c  *     * a
          *       *
        * 3x      *
      * * * * * * *
            b

\(\displaystyle \text{We have: }\;\begin{Bmatrix}\sin3x &=& \frac{a}{c} \\ \\[-3mm] \cos2x &=& \frac{a}{c} \end{Bmatrix}\)

\(\displaystyle \text{Hence, }2x\text{ and }3x\text{ are complements: }\;3x + 2x \:=\:90^o\)


\(\displaystyle \text{Therefore: }\:5x \:=\:90^o \quad\Rightarrow\quad x \:=\:18^o\)

 
ChrisieB said:
hey guys!! i need help badly!!
i 'have exams coming up and the teacher has given us these revision questions but i dont understand them, i've completed the whole chapter except these ones are at the very end and are the very hardest
can someone help?

21. let S=1+2cosx+2cos2x+2cos3x

a) show that S (sin/2x) = sin(7x/2) <<<<< What does that mean?
b) hence show that if x = 2(pie)/7, then 1+2cosx+2cos2x+2cos3x=0

c) by writing S in terms of cosx, prove that cos(2(pie)/7) is a solution to the plynomial equation 8x^3+4x^2-4x-1=0

22.
A) solve sin3x = cos2x, for 0<x<360

a) using solutions of cosa = cosb

B) use compound angle formulae in the above equation to find the exact values of sin18 and sin54
(Hint: 4u^3 - 2u^2-3u+1 = (u-1)(4u^2+2u-1)

C) use results from B to find the exact values of sin, cos and tan of 18 , 36, 54 and 72 degrees

thanks so much to ANYONE who can help me!!!
Click to expand...
 
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