SOLUTION: What is the value of cos(2α) , if sin(3α) =2 sin(α)?

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Question 1126750: What is the value of cos(2α) , if sin(3α) =2 sin(α)?
Found 2 solutions by MathLover1, ikleyn:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

What is the value of cos%282alpha%29 , if

sin%283alpha%29+=2+sin%28alpha%29?

use the following identity:
sin%283alpha%29+=+3+sin%28alpha%29+cos%5E2%28alpha%29+-+sin%5E3%28alpha%29
factor it:
sin%283alpha%29+=sin%28alpha%29%28+3+cos%5E2%28alpha%29+-+sin%5E2%28alpha%29%29
use identity sin%5E2%28alpha%29=1-cos%5E2%28alpha%29

sin%283alpha%29+=sin%28alpha%29%28+3+cos%5E2%28alpha%29+-+%281-cos%5E2%28alpha%29%29+%29
sin%283alpha%29+=sin%28alpha%29%28+3+cos%5E2%28alpha%29+-+1%2Bcos%5E2%28alpha%29%29+%29
sin%283alpha%29+=sin%28alpha%29%28+4cos%5E2%28alpha%29+-+1+%29
then we have
sin%28alpha%29%28+4cos%5E2%28alpha%29+-+1+%29=2sin%28alpha%29
cross%28sin%28alpha%29%29%28+4cos%5E2%28alpha%29+-+1+%29=2cross%28sin%28alpha%29%29
4cos%5E2%28alpha%29+-+1+%29=2...simplify, divide by+2
2cos%282+alpha%29+-+1=1
2cos%282alpha%29+=2
+cos%282alpha%29+=1


Answer by ikleyn(52787) About Me  (Show Source):
You can put this solution on YOUR website!
.

                The solution by the tutor @MathLover1 is not exactly correct,

                so I came to fix it. 


Start from this basic identity


    sin(3a) = 3*sin(a) - 4*sin^3(a).


We are given that


    3*sin(a) - 4*sin^3(a) = 2*sin(a).


It implies


    %284%2Asin%5E2%28a%29+-+1%29%2Asin%28a%29 = 0.


So, either  sin(a) = 0   or   4%2Asin%5E2%28a%29+-+1 = 0;  the last is equivalent to  sin%5E2%28a%29 = 1%2F4.


Case 1.  If  sin(a) = 0,  then  cos(2a) = 1+-+2%2Asin%5E2%28a%29 = 1.


Case 2.  If  sin%5E2%28a%29 = 1%2F4,  then  cos(2a) = 1+-+2%2Asin%5E2%28a%29%29 = 1+-+2%2A%281%2F4%29 = 1%2F2.


Answer.  If  sin(3a) =2*sin(a),  then EITHER  cos(2a) = 1  OR  cos(2a) = 1%2F2.

Solved.