Find two angles in the interval [0°, 360°) that satisfy each of the following. Round answers to the nearest degree.sin θ = 0.52991926

Verified step by step guidance

1. Step 1: Understand the problem. We need to find two angles \( \theta \) in the interval [0°, 360°) such that \( \sin \theta = 0.52991926 \).
2. Step 2: Use the inverse sine function to find the principal angle. Calculate \( \theta_1 = \sin^{-1}(0.52991926) \) to find the first angle.
3. Step 3: Determine the second angle. Since sine is positive in the first and second quadrants, the second angle \( \theta_2 \) can be found using \( \theta_2 = 180° - \theta_1 \).
4. Step 4: Verify that both angles are within the interval [0°, 360°). If any angle is outside this range, adjust it by adding or subtracting 360° as necessary.
5. Step 5: Round both angles to the nearest degree as required by the problem statement.

Key Concepts

Sine Function

The sine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is periodic with a range of [-1, 1] and is defined for all real numbers. Understanding the sine function is crucial for solving equations involving angles, as it helps determine the values of angles that yield specific sine values.

Inverse Sine (Arcsin)

The inverse sine function, or arcsin, is used to find an angle when the sine value is known. It takes a value from the range [-1, 1] and returns an angle in the interval [-90°, 90°]. For angles outside this range, additional solutions can be found using the periodic nature of the sine function, which is essential for determining all possible angles that satisfy the given sine equation.

Angle Solutions in Trigonometry

In trigonometry, angles can have multiple solutions due to the periodic nature of trigonometric functions. For sine, if sin θ = k, the general solutions can be expressed as θ = arcsin(k) + 360°n and θ = 180° - arcsin(k) + 360°n, where n is any integer. This concept is vital for finding all angles within a specified interval, such as [0°, 360°), ensuring that all valid solutions are considered.