Solve the following equations.


{Use of Tech} $\(\sin\)2\(\theta\)=\(\frac\)15,0\(\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.

Key Concepts

Trigonometric Functions

Trigonometric functions, such as sine, cosine, and tangent, relate the angles of a triangle to the lengths of its sides. The sine function, specifically, gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. Understanding these functions is crucial for solving equations involving angles, particularly in the context of periodic functions and their properties.

Double Angle Formulas

Double angle formulas are identities that express trigonometric functions of double angles in terms of single angles. For example, the sine double angle formula states that sin(2θ) = 2sin(θ)cos(θ). This concept is essential for simplifying and solving equations that involve angles multiplied by two, as seen in the given equation.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In the context of the given problem, the interval 0 < θ < π/2 indicates that θ must be a positive angle less than 90 degrees. Understanding interval notation is important for determining the valid solutions to trigonometric equations, ensuring that the solutions fall within specified bounds.