Question

For what values of `theta` does the equation `2x^2+(4sintheta)x-cos⁡(2theta)=0` have distinct real roots?

I still don't understand the concept of sin and tan, as well as theta. Any help is very much appreciated and I thank you for any time taken to solve and assist!

Answer 1

any real value of `\theta` i.e. `\theta\in\mathbbR`

Explanation:

The given quadratic equation: `2x^2+(4\sin\theta)x-\cos2\theta=0` will have real & distinct roots if its discriminant `B^2-4AC` is positive

`B^2-4AC>0`

`(4\sin\theta)^2-4(2)(-\cos2\theta)>0`

`16\sin^2\theta+8\cos2\theta>0`

`16\sin^2\theta+8(1-2\sin^2\theta)>0`

`16\sin^2\theta+8-16\sin^2\theta>0`

`8>0`

The above result is independent of `\theta` i.e. it is true for all values of `\theta`

`\therefore \theta\in\mathbbR`