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If sec(x) + tan(x) = 4/3, then find sin(x).

 Jun 28, 2022
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sec x  + tan x =   4/3

 

[ Note that   tan^2 x + 1  =  sec^2x    →   tan^2 x =  sec ^2 x - 1   →  tan x = sqrt ( sec^2 x - 1)  ] 

 

So

 

sec x +  sqrt  [sec^2 x - 1 ] =  4/3

 

sqrt [ sec^2 x - 1 ]=  4/3  - secx          square both sides

 

sec^2 x  - 1  =   16/9 - (8/3)sec x + sec^2 x       simplify

 

-1  = 16/9 - (8/3)sec x     rearrange as

 

(8/3)sec x  = 16/9 + 1

 

(8/3)sec x =  25 / 9

 

sec x = (25/9)(3/8)  = 75/72  =  25/24

 

cos x=  24/25

 

sin x = sqrt [  25^2 - 24^2 ]  / 25  =  sqrt [ 49] / 25 =  7 / 25

 

 

cool cool cool 

 Jun 28, 2022

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