Question

solve the inequality cos2x+9sinx-5=0​

Answer 1

Answer:

30°

Step-by-step explanation:

To Solve :-

cos2x + 9sinx - 5 = 0

How To Do :-

First of all We need to change 'cos2x' in terms of 'sinx' , after changing that we can observe that all terms are with same variable(sinx) so we need to find the value of sinx by splitting the middle term then we can see that only one value of sinx satisfying the range of it. So we can conclude it is only the value of 'x'

Formula Required :-

cos2x = 1 - 2sin²x

Solution :-

cos2x + 9sinx - 5 = 0

1 - 2sin²x + 9sinx - 5 = 0

- 2sin²x + 9sinx - 5 + 1 = 0

- 2sin²x + 9sinx - 4 = 0

Taking '-' common :-

-(2sin²x - 9sinx + 4) = 0

2sin²x - 9sinx + 4 = 0

Splitting the middle term :

2sin²x - 8sinx - sinx + 4 = 0

2sinx(sinx - 4) - 1 (sinx - 4) = 0

(sinx - 4) (2sinx - 1) = 0

Equating both terms to '0' :-

sinx - 4 = 0 , 2sinx - 1 = 0

Solving for sinx -4 = 0

sinx = 4

sinx ≠ 4

[ ∴ Range of sinx is in between - 1 and 1]

Solving for '2sinx - 1 = 0'

2sinx = 1

sinx = 1/2

x = sin⁻¹(1/2)

x = sin⁻¹(sin30°)

x = 30°

∴ Value of 'x' = 30°