Find a value of θ in the interval [0°, 90°) that satisfies each statement. Give answers in decimal degrees to six decimal places. See Example 2.sin θ = 0.84802194

Verified step by step guidance

1. Step 1: Understand that you need to find an angle \( \theta \) such that \( \sin \theta = 0.84802194 \).
2. Step 2: Use the inverse sine function, also known as arcsin, to find \( \theta \). This is expressed as \( \theta = \sin^{-1}(0.84802194) \).
3. Step 3: Ensure your calculator is set to degree mode since the problem asks for the answer in decimal degrees.
4. Step 4: Input the value 0.84802194 into your calculator and apply the inverse sine function to find \( \theta \).
5. Step 5: Verify that the resulting angle \( \theta \) is within the specified interval [0°, 90°).

Key Concepts

Sine Function

The sine function, denoted as sin(θ), is a fundamental trigonometric function that relates the angle θ in a right triangle to the ratio of the length of the opposite side to the hypotenuse. It is defined for all angles and is periodic, with a range of values between -1 and 1. Understanding the sine function is crucial for solving problems involving angles and their corresponding ratios.

Inverse Sine Function

The inverse sine function, or arcsin, is used to find the angle θ when the sine value is known. It is denoted as sin⁻¹(x) and returns an angle in the range of [-90°, 90°]. In this context, using the inverse sine function allows us to determine the angle that corresponds to the given sine value of 0.84802194 within the specified interval.

Angle Measurement in Degrees

Angles can be measured in degrees, with a full circle comprising 360 degrees. In trigonometry, it is common to express angles in degrees, especially in practical applications. When solving for θ in the interval [0°, 90°), it is important to provide the answer in decimal degrees, ensuring precision up to six decimal places as specified in the question.