Domanda del libro di testo
In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). cos² x - cos x - 1 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Capitolo 3, Problema 85
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
Guida verificata passo dopo passo1
Identify the equation to solve: \(\sin x = 0.8246\) on the interval \([0, 2\pi)\).
Use the inverse sine function to find the principal solution: \(x = \sin^{-1}(0.8246)\).
Calculate the principal value using a calculator, ensuring the mode is set to radians.
Recall that sine is positive in the first and second quadrants, so find the second solution using \(x = \pi - \sin^{-1}(0.8246)\).
List both solutions within the interval \([0, 2\pi)\) and express them rounded to four decimal places.
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Solving Trigonometric Equations
Solving trigonometric equations involves finding all angle values within a specified interval that satisfy the given equation. For sine equations, this means identifying angles whose sine value matches the given number, considering the periodic nature of the sine function.
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Percorso guidato
4:34
How to Solve Linear Trigonometric Equations
Using the Inverse Sine Function
The inverse sine function (sin⁻¹ or arcsin) is used to find the principal angle whose sine is a given value. Since sine is positive in the first and second quadrants, two solutions typically exist within [0, 2π), which must be calculated and verified.
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Percorso guidato
4:03Inverse Sine
Interval and Periodicity of Sine Function
The sine function has a period of 2π, meaning its values repeat every 2π radians. When solving on the interval [0, 2π), it is important to find all solutions within one full cycle, including angles in both the first and second quadrants where sine is positive.
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5:33Period of Sine and Cosine Functions
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