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

Verified step by step guidance

1. Start with the given equation: \(5 \cot^{2} x - 15 = 0\).
2. Isolate the \(\cot^{2} x\) term by adding 15 to both sides and then dividing by 5: \(\cot^{2} x = \frac{15}{5}\).
3. Simplify the right side to get \(\cot^{2} x = 3\).
4. Take the square root of both sides to solve for \(\cot x\): \(\cot x = \pm \sqrt{3}\).
5. 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.

Key Concepts

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.

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).

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.