SOLUTION: Determine the exact value of x. In other words, your answer must not use decimals cos⁡2x-sin⁡x=0

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Question 1110695: Determine the exact value of x. In other words, your answer must not use decimals cos⁡2x-sin⁡x=0
Found 2 solutions by greenestamps, Fombitz:
Answer by greenestamps(13195) About Me  (Show Source):
You can put this solution on YOUR website!


You have three identities for cos(2x):
%28cos%28x%29%29%5E2-%28sin%28x%29%29%5E2
2%28cos%28x%29%29%5E2-1
1-2%28sin%28x%29%29%5E2

Since the other term in the equation is sin(x), you want to use the identity for cos(2x) that contains only sin(x); that will give you a quadratic equation in sin(x). So

cos%282x%29-sin%28x%29+=+0
1-2%28sin%28x%29%29%5E2-sin%28x%29+=+0
-2%28sin%28x%29%29%5E2-sin%28x%29%2B1+=+0
2%28sin%28x%29%29%5E2%2Bsin%28x%29-1+=+0
%282%28sin%28x%29%29-1%29%28sin%28x%29%2B1%29+=+0
sin%28x%29+=+1%2F2 or sin%28x%29+=+-1

The wording of the question is strange: "Determine the exact value of x." There are an infinite number of values of x that satisfy the equation....

For solutions on the interval [0,2pi), the solutions are {pi/6, 5pi6, 3pi/2}.
A graph:

graph%28400%2C400%2C0%2C2pi%2C-2%2C2%2Ccos%282x%29-sin%28x%29%29

Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
cos%282x%29-sin%28x%29=0
1-2sin%5E2%28x%29-sin%28x%29=0
2sin%5E2%28x%29%2Bsin%28x%29-1=0
Use a substitution,
u=sin%28x%29
So,
2u%5E2%2Bu-1=0
%28u%2B1%29%282u-1%29=0
Two "u" solutions:
u%2B1=0
u=-1
sin%28x%29=-1
x=180
and
2u-1=0
2u=1
u=1%2F2
sin%28x%29=1%2F2
x=30 and x=150