Solve the following equations.


{Use of Tech} $\sin2\theta=\frac15,0\lt{\theta}\lt{\frac{\pi}{2}}$

Verified step by step guidance

1. First, recognize that the equation given is \( \sin 2\theta = \frac{1}{5} \). This is a trigonometric equation involving the sine function.
2. Next, use the inverse sine function to solve for \( 2\theta \). This gives \( 2\theta = \arcsin\left(\frac{1}{5}\right) \).
3. Since the sine function is periodic with a period of \( 2\pi \), consider the general solution for \( 2\theta \), which is \( 2\theta = \arcsin\left(\frac{1}{5}\right) + 2k\pi \) or \( 2\theta = \pi - \arcsin\left(\frac{1}{5}\right) + 2k\pi \), where \( k \) is an integer.
4. Now, solve for \( \theta \) by dividing the entire equation by 2, giving \( \theta = \frac{1}{2}\arcsin\left(\frac{1}{5}\right) + k\pi \) or \( \theta = \frac{1}{2}(\pi - \arcsin\left(\frac{1}{5}\right)) + k\pi \).
5. Finally, apply the constraint \( 0 < \theta < \frac{\pi}{2} \) to find the specific values of \( \theta \) that satisfy the original equation within the given interval. Check each possible solution to ensure it falls within this range.