If 2 SinA + 3 CosA = 2 , then 3 SinA - 2 CosA = ?
Answers
Answer:
±3
Step-by-step explanation:
Given,
2sinA + 3cosA = 2
To Find :-
Value of :-
3sinA - 2cosA?
How To Do :-
As they given the value of '2sinA + 3cosA' we need to consider that as an equation and we need to do squaring on both sides for that equation and we need to find the value of 'cosA' we can observe that there are '2' different values of 'cosA' after obtaining that we need to find that values of 'sinA' and we need to find the required.
Solution :-
Taking the given condition :-
2sinA + 3cosA = 2
Squaring on both sides :-
(2sinA + 3cosA)² = (2)²
(2sinA)² + (3cosA)² + 2(2sinA)(3cosA) = 4
4sin²A + 9cos²A + 12sinAcosA = 4
12sinAcosA = 4 - 4sin²A - 9cos²A
12sinAcosA = 4(1 - sin²A) - 9cos²A
12sinAcosA = 4(cos²A) - 9cos²A
[ ∴ cos²A = 1 - sin²A ]
12sinAcosA = 4cos²A - 9cos²A
12sinAcosA = - 5cos²A
12sinAcosA + 5cos²A = 0
Taking 'cosA' as common :-
cosA(12sinA + 5cosA) = 0
Equating both terms to 'o' :-
→ cosA = 0 , 12sinA + 5cosA = 0
cosA = 0 , 5cosA = - 12sinA
cosA = 0 , cosA = -12sinA/5
If cosA = 0
Substituting the value of cosA in '2sinA + 3cosA = 2' to get the value of 'sinA' :-
2sinA + 3(0) = 2
2sinA + 0 = 2
2sinA = 2
sinA = 2/2
sinA = 1
3sinA - 2cosA
= 3(1) - 2(0)
= 3 - 0
= 3
∴ 3sinA - 2cosA = 3.
if 'cosA = -12sinA/5' :-
Substituting the value of cosA in '2sinA + 3cosA = 2' to get the value of 'sinA' :-
2sinA + 3(-12sinA)/5 = 2
2sinA - 36sinA/5 = 2
taking L.C.M :-
5(2sinA) - 36sinA/5 = 2
10sinA - 36sinA = 2 × 5
-26sinA = 10
sinA = 10/-26
sinA = 5/-13
sinA = -5/13
→ cosA = -12sinA/5
= -12(-5/13)/5
= 60/13 × 1/5
= 12/13
3sinA - 2cosA :-
= 3(-5/13) - 2(12/13)
= -15/13 - 24/13
= -15 - 24/13
= -39/13
= - 3
∴ 3sinA - 2cosA = - 3
Since we can observe that there are two different values of '3sinA - 2cosA' :-
→ 3sinA - 2cosA = ±3