SOLUTION: If 2 cos@ + sec@=3 then find the value of @.

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Question 830307: If 2 cos@ + sec@=3 then find the value of @.
Found 2 solutions by Fombitz, Edwin McCravy:
Answer by Fombitz(32388) About Me  (Show Source):
You can put this solution on YOUR website!
2%2Acos%28A%29%2Bsec%28A%29=3
2%2Acos%28A%29%2B1%2Fcos%28A%29=3
2%2Acos%28A%29%5E2%2B1=3%2Acos%28A%29
Let u=cos%28A%29
2u%5E2-3u%2B1=0
%28u-1%29%282u-1%29=0
Two "u" solutions:
u-1=0
u=1
cos%28A%29=1
highlight%28A=0%29
and
2u-1=0
2u=1
u=1%2F2
cos%28A%29=1%2F2
Oops! I screwed up the previous solution.
highlight%28A=60%29 and highlight%28A=300%29
Three solutions.
Oops! I screwed up the previous solution.
A can be 0, 60, and 300 degrees.

Answer by Edwin McCravy(20060) About Me  (Show Source):
You can put this solution on YOUR website!
2cos%28theta%29+%2B+sec%28theta%29%22%22=%22%223

Replace sec%28theta%29 by 1%2Fcos%28theta%29

2cos%28theta%29+%2B+1%2Fcos%28theta%29%22%22=%22%223

Multiply through by cos%28theta%29

2cos%5E2%28theta%29+%2B+1%22%22=%22%223cos%28theta%29

Get 0 on the right side by adding -cos%28theta%29 to both sides:

2cos%5E2%28theta%29+-+3cos%28theta%29%2B1%22%22=%22%22%220%22

Just as 2x²-3x+1 factors as (x-1)(2x-1), the above factors as:

%28cos%28theta%29-1%29%282cos%28theta%29-1%29%22%22=%22%22%220%22

We use the zero-factor property and set each factor = 0

cos%28theta%29-1%22%22=%22%22%220%22,    2cos%28theta%29-1%22%22=%22%22%220%22 

cos%28theta%29%22%22=%22%221,    cos%28theta%29%22%22=%22%221%2F2
                                                    
theta=0,     theta=pi%2F3,5pi%2F3

or any positive or negative multiple of 2pi added to these.

or in degrees, 0°, 60°, 300°

or any positive or negative multiple of 360° added to these.

Edwin