Domanda del libro di testo
Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 9 tan² x - 3 = 0
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Ch. 3 - Trigonometric Identities and Equations
Blitzer3rd EditionTrigonometryISBN: 9780137316601
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Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.5.43
Blitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 3.5.43Capitolo 3, Problema 3.5.43
Exercises 39–52 involve trigonometric equations quadratic in form. Solve each equation on the interval [0, 2𝝅). 2 sin² x = sin x + 3
Guida verificata passo dopo passo1
Rewrite the given equation to standard quadratic form by bringing all terms to one side: \(2 \sin^{2} x - \sin x - 3 = 0\).
Let \(u = \sin x\) to transform the trigonometric equation into a quadratic equation in terms of \(u\): \(2u^{2} - u - 3 = 0\).
Solve the quadratic equation \(2u^{2} - u - 3 = 0\) using the quadratic formula \(u = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=2\), \(b=-1\), and \(c=-3\).
Find the values of \(u\) (which represent \(\sin x\)) from the quadratic solutions and determine which values are valid since \(\sin x\) must be in the interval \([-1, 1]\).
For each valid \(u\) value, solve for \(x\) in the interval \([0, 2\pi)\) by using the inverse sine function and considering the sine function's periodicity and symmetry.
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Quadratic Form in Trigonometric Equations
Some trigonometric equations can be rewritten to resemble quadratic equations by expressing terms like sin²x or cos²x as a single variable squared. This allows the use of algebraic methods such as factoring or the quadratic formula to find solutions for the trigonometric function.
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Percorso guidato
5:35
Introduction to Quadratic Equations
Solving Trigonometric Equations on a Specific Interval
When solving trigonometric equations, it is important to find all solutions within the given interval, here [0, 2π). This involves determining all angle values that satisfy the equation within one full cycle of the sine or cosine function.
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4:34How to Solve Linear Trigonometric Equations
Using the Unit Circle to Find Angle Solutions
The unit circle provides a geometric interpretation of sine and cosine values for angles between 0 and 2π. By knowing the sine values corresponding to specific angles, one can identify all solutions to the equation within the interval.
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06:11Introduction to the Unit Circle
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