SOLUTION: If (sec x-2)(2sec x-1)=0

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Question 80734: If (sec x-2)(2sec x-1)=0
Found 2 solutions by stanbon, jim_thompson5910:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
(sec x-2)(2sec x-1)=0
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2sec^2x-3secx +2=0
(2secx+1))(secx-2)=0
secx= -1/2 or secx = 2
Since sec has no value between -1 and 1 only secx=2 is a viable answer:
Therefore : secx=2 or cosx=1/2:
and x = 60 degrees or 300 degrees.
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Cheers,
Stan H.

Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!
%28sec+x-2%29%282sec+x-1%29=0

2sec%5E2%28x%29-sec%28x%29-4sec%28x%29%2B2=0 Foil

2sec%5E2%28x%29-5sec%28x%29%2B2=0 Combine like terms
Let y=sec%28x%29
2y%5E2-5y%2B2=0
Now factor to solve for y (note: this solver uses x instead of y):

ERROR Algebra::Solver::Engine::invoke_solver_noengine: solver not defined for name 'solving_by_factoring'.
Error occurred executing solver 'solving_by_factoring' .



Since we have the solutions

y=1%2F2 or y=2

we can say

sec%28x%29=1%2F2 or sec%28x%29=2

which is equivalent to

1%2Fcos%28x%29=1%2F2 or 1%2Fcos%28x%29=2

Now invert both sides

cos%28x%29=2%2F1 or cos%28x%29=1%2F2

Now take the arccosine of both sides to solve for x


x=cos%5E%28-1%29%282%2F1%29 or x=cos%5E%28-1%29%281%2F2%29

Since the arccosine of 2 is undefined we must discard that answer. So our answer is
x=pi%2F3%2Bpi%2An where n is an integer
or
x=-pi%2F3%2Bpi%2An where n is an integer

which is equivalent in degrees:

x=60%2B360%2An where n is an integer
or
x=-60%2B360%2An where n is an integer