In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅).sin x = 0.8246

Verified step by step guidance

1. Identify the equation: \( \sin x = 0.8246 \).
2. Use the inverse sine function to find the principal value: \( x = \sin^{-1}(0.8246) \).
3. Calculate the principal value using a calculator, ensuring the result is in radians.
4. Determine the second solution in the interval \([0, 2\pi)\) by using the identity \( x = \pi - \sin^{-1}(0.8246) \).
5. Verify both solutions are within the interval \([0, 2\pi)\) and round them to four decimal places.

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 sine, as it helps identify possible angles that yield specific sine values.

Inverse Sine (Arcsin)

The inverse sine function, denoted as arcsin or sin⁻¹, is used to find the angle whose sine is a given value. It returns values in the range of [-π/2, π/2]. When solving equations like sin x = 0.8246, using the inverse sine allows us to determine the principal angle, which is the first step in finding all solutions within a specified interval.

Periodic Nature of Trigonometric Functions

Trigonometric functions, including sine, are periodic, meaning they repeat their values in regular intervals. For the sine function, the period is 2π, which implies that if sin x = k for some value k, then sin(x + 2nπ) = k for any integer n. This property is essential when solving equations over a specified interval, as it allows us to find all possible solutions by considering the periodicity of the sine function.