Solving a sin &#x2061;<!-- ⁡ --> x + b cos &#x2061;<!-- ⁡ --> x = c Suppose

pipantasi4 2022-06-30 Answered
Solving a sin x + b cos x = c
Suppose a sin x + b cos x = c . My teacher told me that with changing variable we can solve it by this : ( c + b ) t 2 2 a t + ( c b ) = 0 where t = tan x 2 . I thought it is true always but today found a problem . When we define sin x and cos x with tan x 2 , we should consider cos x 2 0 and x 2 k π + π but it can be the answer of the main equation (i.e. a sin x + b cos x = c ) ! So , that formula is incomplete ?
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Answers (1)

Maggie Bowman
Answered 2022-07-01 Author has 14 answers
use that
a 2 + b 2 ( a a 2 + b 2 sin ( x ) + b a 2 + b 2 cos ( x ) ) = c
and
cos ( ϕ ) = a a 2 + b 2
sin ( ϕ ) = b a 2 + b 2
therefore we get
sin ( x + ϕ ) = c a 2 + b 2
or you write
2 a tan ( x / 2 ) 1 + ( tan ( x / 2 ) ) 2 + b ( 1 ( tan ( x / 2 ) ) 2 ) 1 + ( tan ( x / 2 ) ) 2 = c
with
tan ( x / 2 ) = t
as your teacher said
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