# Trig equation involving multiple angles

1. Apr 21, 2013

### waterliyl

Question

Find all the values of ∏ in the interval 0 < θ < 2∏ for which
2cos2 θ – sin θ = 1,
giving each answer in terms of ∏

I think the question is asking me to find the values of θ in terms of ∏ for all values between 0 < θ < 2∏

This is what I have done so far
I have worked out this question to give me a quadratic in terms of sin :

2sin2 θ + sin θ – 1 = 0

sinθ = 1/2 or -1

so θ = ∏/6 or -∏/2

3. The attempt at a solution

Now, I have to find the values of ∏ within a range.
The method I was taught at school was
PV2 = 2∏ - PV1

Which gave me: 5∏/6 and 3∏/2

For the rest of the values, ± ∏

So I did that giving me 11∏/6, 5∏/2

Note: -∏/2 isn't part of the solution as it is out of the range

So I think the final solutions are: ∏/6, 5∏/6, 3∏/2, 11∏/6, 5∏/2

but my markscheme is saying that the solutions are ∏/6, 5∏/6, 3∏/2

so I don't know what I have done wrong? Why should I not ±∏ from my PV 1 and 2 on this occasion?
Am I doing something wrong?

Note: I know there is some CAST method, but I haven't been taught this so I don't know how to use it.

Thank you so much for your time and help!
I really appreciate it! :)

Last edited: Apr 21, 2013
2. Apr 21, 2013

### Mandelbroth

$\displaystyle \forall \theta, \pi=4\int_{0}^{1} \sqrt{1-x^2}dx$. Clarify your question, or it will not be answered.