IF 3sinA + 5cosA = 5 PROVE THAT: 5sinA - 3cosA = plus or minus 3 and HENCE FIND (3 - 5tanA)/(5 + 3tanA)

Dear Student,

 

Given equation is

 

3sinA + 5cosA = 5

 

=> 3sinA = 5(1 - cosA)

 

=> 9sin²A = 25(1 + cos²A - 2cosA)

 

=> 9(1-cos²A) = 25 + 25cos²A - 50cosA

 

=> 34cos²A - 50cosA + 8 = 0

 

solving the quadratic equation, we get

 

cosA = 1 or cosA = 8/17

 

=> sinA = 0 or sinA = 15/17

 

 

Hence, when cosA = 1, we have

 

5sinA - 3cosA = 5 x 0 - 3 x 1 = -3

 

and when cosA = 8/17, we have

 

5 x 15/17 - 3 x 8/17 = (75 - 24)/17 = 51/17 = 3

 

Therefore, 5sinA - 3cosA = 3 or -3.

 

Regards Topperlearning.