Find all solutions of each equation. 2 cos x + √ 3 = 0

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Start by isolating the cosine term in the equation: \(2 \cos x + \sqrt{3} = 0\). Subtract \(\sqrt{3}\) from both sides to get \(2 \cos x = -\sqrt{3}\).

Next, divide both sides of the equation by 2 to solve for \(\cos x\): \(\cos x = -\frac{\sqrt{3}}{2}\).

Recall the unit circle values where \(\cos x = -\frac{\sqrt{3}}{2}\). Cosine is negative in the second and third quadrants. Identify the reference angle where \(\cos x = \frac{\sqrt{3}}{2}\), which is \(\frac{\pi}{6}\).

Use the reference angle to find the general solutions in the second and third quadrants: \(x = \pi - \frac{\pi}{6}\) and \(x = \pi + \frac{\pi}{6}\).

Write the general solution for all angles by adding the period of cosine, \(2\pi k\), where \(k\) is any integer: \(x = \pi \pm \frac{\pi}{6} + 2\pi k\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Solving Basic Trigonometric Equations

This involves isolating the trigonometric function (e.g., cosine) and finding all angles that satisfy the equation within a given domain. For example, solving 2 cos x + √3 = 0 requires rewriting it as cos x = -√3/2 and then determining all x values where cosine equals this value.

Unit Circle and Reference Angles

The unit circle helps identify angles corresponding to specific cosine values. Reference angles are acute angles used to find solutions in different quadrants. Knowing that cos x = -√3/2 corresponds to angles in the second and third quadrants is essential for finding all solutions.

General Solution for Trigonometric Equations

Trigonometric functions are periodic, so solutions repeat at regular intervals. The general solution for cos x = a is x = ± arccos(a) + 2πn, where n is any integer. This formula ensures all possible solutions are captured, not just those in one cycle.

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