Question

How do you solve `12sinx-7sqrtsinx+1=0` in the interval [0,360]?

Answer 1

Start by isolating the square root.

`12sinx +1 = 7sqrt(sinx)`

`(12sinx + 1)^2 = (7sqrt(sinx))^2`

`144sin^2x + 24sinx + 1 = 49sinx`

`144sin^2x - 25sinx + 1 = 0`

We let `t= sinx`:

`144t^2 - 25t + 1 = 0`

Solve by factoring:

`144t^2 - 16t - 9t + 1 = 0`

`16t(9t - 1) - 1(9t - 1) = 0`

`(16t - 1)(9t - 1) = 0`

`t = 1/16 and 1/9`

`sinx = 1/16 and sinx = 1/9`

`x = arcsin(1/16), 180˚ - arcsin(1/16), arcsin(1/9) and 180˚ - arcsin(1/9)`

Make sure your calculator is in degree mode when approximating these values.

After checking your answers for extraneous solutions, you should get that all solutions work.

Hopefully this helps!