General solution of 2cosx squared x +5sinx=4
Answers
Answer:-
sin x = 1/2
Given to find the solution :-
2cos²x + 5sinx = 4
Solution :-
Firstly lets convert the entire equation in terms of the trigonometric ratio "sin"
From trigonometric identities ,
sin²A +cos²A = 1
cos²A = 1-sin²A
Substitute this value in given equation .Since , the entire equation will be converted in terms of "sin"
= 2( 1-sin²x ) +5sinx = 4
= 2- 2sin² x + 5 sin x - 4 =0
= -2sin²x +5 sin x -2 =0
Take common " - "
= 2sin²x - 5sinx +2 =0
Now we have the Quadratic equation 2sin²x -5sinx +2 =0
Finding the factors through splitting the middle term
2sin²x -5sinx +2 =0
2sin²x - sin x - 4sinx +2 =0
sin x (2sinx -1) -2 (2sinx -1) =0
(2sinx -1)( sinx-2) =0
2sinx -1 =0 and sin x -2 =0
2sinx = 1
sin x =1/2
sin x -2 =0
sin x =2
But As we know the range of sin A that is
-1≤sinA ≤ 1
So,
sin x ∉ 2
sin x ∈ 1/2
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Know more :-
Range of cos A is -1≤cosA ≤ 1
Range of sec A is sec A≤ -1 and sec A ≥ 1
Range of csc A is csc A≤ -1 and csc A ≥ 1