Final answer:
To solve the equation cos(2\theta) = sin(\theta), use the trigonometric identity cos(2\theta) = 1 - 2sin^2(\theta), rearrange into a quadratic equation of sin(\theta), and then solve to get the values of \theta within the range 0 < \theta < 2\pi.
Step-by-step explanation:
To solve the equation cos(2\theta) = sin(\theta) for 0 < \theta < 2\pi, we can use trigonometric identities. From identity 14 in the provided references, we know that:
- cos(2\theta) = 1 - 2sin2(\theta)
Substituting this into our original equation gives:
1 - 2sin2(\theta) = sin(\theta)
Let's rearrange the equation and solve for sin(\theta):
-2sin2(\theta) - sin(\theta) + 1 = 0
This is a quadratic equation in terms of sin(\theta), and we can use the quadratic formula to find the solutions:
sin(\theta) = [-(-1) ± \sqrt{(-1)2 - 4(-2)(1)}] / (2 * -2)
sin(\theta) = (1 ± \sqrt{1 + 8}) / -4
sin(\theta) = (1 ± 3) / -4
Therefore, sin(\theta) has two possible values:
- sin(\theta) = -1/2
- sin(\theta) = -1
We then need to consider the range 0 < \theta < 2\pi and check which of these solutions are valid within this range.