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Ch. 3 - Trigonometric Identities and Equations
Blitzer - Trigonometry 3rd Edition
Blitzer3rd EditionTrigonometryISBN: 9780137316601Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 3rd EditionCh. 3 - Trigonometric Identities and EquationsProblema 109
Capitolo 3, Problema 109

In Exercises 97–116, use the most appropriate method to solve each equation on the interval [0, 2𝝅). Use exact values where possible or give approximate solutions correct to four decimal places. 5 cot² x - 15 = 0

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Start with the given equation: \(5 \cot^{2} x - 15 = 0\).
Isolate the \(\cot^{2} x\) term by adding 15 to both sides and then dividing by 5: \(\cot^{2} x = \frac{15}{5}\).
Simplify the right side to get \(\cot^{2} x = 3\).
Take the square root of both sides to solve for \(\cot x\): \(\cot x = \pm \sqrt{3}\).
Recall that \(\cot x = \frac{\cos x}{\sin x}\), and use this to find the values of \(x\) in the interval \([0, 2\pi)\) where \(\cot x = \sqrt{3}\) and \(\cot x = -\sqrt{3}\). Consider the unit circle and the signs of sine and cosine in each quadrant to determine the exact or approximate solutions.

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Trigonometric Equations

Trigonometric equations involve functions like sine, cosine, and cotangent. Solving these equations means finding all angle values within a given interval that satisfy the equation. Understanding how to manipulate and isolate trigonometric functions is essential for finding exact or approximate solutions.
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Cotangent Function and Its Properties

The cotangent function, cot(x), is the reciprocal of the tangent function, defined as cot(x) = cos(x)/sin(x). It is periodic with period π and undefined where sin(x) = 0. Recognizing its behavior and domain restrictions helps in solving equations involving cot²(x).
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Solving Quadratic Equations in Trigonometric Form

Equations like 5 cot² x - 15 = 0 can be treated as quadratic in cot(x). By isolating cot² x and taking square roots, you find possible values for cot x. Then, using inverse trigonometric functions and considering the interval, you determine all valid solutions.
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