Question

Find all solutions to the equation. 7 sin2x - 14 sin x 2 = -5

Answer 1

The all solution of given trigonometrical equation is
`(\Pi )/(2)` + 2πn .

Explanation:

Given trigonometrical equation as :

7
`sin^(2)x` - 14 sin x + 2 = - 5

Now, Rearranging the equation

7
`sin^(2)x` - 14 sin x + 2 + 5 = 0

Or, 7
`sin^(2)x` - 14 sin x + 7 = 0.

Taking 7 as common

So,
`sin^(2)x` - 2 sin x + 1 = 0.

∵The standard form of quadratic equation

a x² + b x + c = 0

So, breaking the mid term we get

`sin^(2)x` - sin x - sin x + 1 = 0.

Or, sin x (sin x - 1) - 1 (sin x - 1) = 0

Or, (sin x - 1) (sin x - 1) = 0

Or, (sin x -1) = 0 and (sin x - 1) = 0

So, sin x = 1

∴ x =
`sin^(-1)1`

i.e x = 90°

Or, ∠x =
`(\Pi )/(2)`

And for all solution
`(\Pi )/(2)` + 2πn , where n∈ 0 to ....

Hence, The all solution of given trigonometrical equation is
`(\Pi )/(2)` + 2πn . Answer