SOLUTION: If sec x= 3/2 and cosec y= 3; determine cos (x+y)

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Question 1197232: If sec x= 3/2 and cosec y= 3; determine cos (x+y)

Found 3 solutions by Alan3354, Theo, ikleyn:
Answer by Alan3354(69443)   (Show Source): You can put this solution on YOUR website!
If sec x= 3/2 and cosec y= 3; determine cos (x+y)
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sec = 3/2 ---> cos(x) = 2/3

----
csc = 3 ----> sin(y) = 1/3

==========================
cos(x+y) = cos(x)*cos(y) - sin(x)*sin(y) ------ I'm pretty sure.
Look it up at "half-angle formulas" on Wikipedia.
Answer by Theo(13342)   (Show Source): You can put this solution on YOUR website!
sec(x) = 3/2 becomes:
1/cos(x) = 3/2 which becomes:
cos(x) = 2/3.
when cos(x) = 2/3, solve for sin(x) to get sin(x) = sqrt(5)/3
confirm by cos^2(x) + sin^2(x) = 1 which becomes:
(2/3)^2 + (sqrt(5)/3)^2 = 1

you have:
cos(x) = 2/3
sin(x) = sqrt(5)/3

cosec(y) = 3 becomes:
1/sin(y) = 3 which becomes:
sin(y) = 1/3
when sin(y) = 1/3, solve for cos(y) to get cos(y) = sqrt(8)/3
confirm by cos^2(y) + sin^2(y) = 1 which becomes:
(sqrt(8)/3)^2 + (1/3)^2 = 1

you have:
cos(y) = sqrt(8)/3
sin(y) = 1/3

put theme together, you have:
cos(x) = 2/3
cos(y) = sqrt(8)/3
sin(x) = sqrt(5)/3
sin(y) = 1/3

cos(x+y) = cos(x)cos(y)-sin(x)sin(y) which becomes
cos(x+y) = 2/3*sqrt(8)/3 - sqrt(5)/3*1/3 which is equal to .3800873636.

i also used my calculator to confirm this is true.
sec(x) = 3/2 becomes 1/cos(x) = 3/2 which becomes cos(x) = 2/3
cosec(y) = 3 becomes 1/sin(y) = 3 which becomes sin(y) = 1/3
arccos(2/3) = 48.1896851 degrees.
arcsin(1/3) = 19.47122063 degrees.
48.1896851 + 19.47122063 = 67.66090574 degrees.
cos(67.66090574) = .3800873636.



Answer by ikleyn(52803)   (Show Source): You can put this solution on YOUR website!
.
If sec x= 3/2 and cosec y= 3; determine cos (x+y)
~~~~~~~~~~~~~~~ 

If sec(x) = 3/2, then cos(x) = 1/sec(x) = 2/3,

    but we don't know if the angle (the argument) x is in QI or in QIV.


    As a consequence, we can not determine sin(x) by a unique way.



If cosec(y) = 3, then sin(y) = 1/cosec(y) = 1/3,

    but we don't know if the angle (the argument) y is in QI or in QII.


    As a consequence, we can not determine cos(y) by a unique way.



THEREFORE, in this problem we can not determine cos(x+y) by a unique way.

        The problem is posed in  EXTREMELY  INNACCURATE  way:  professional Math composers
        NEVER  formulate problems in this way,  respecting their readers.



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