Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.

3 tan 3x = √3

검증된 단계별 안내

Start by isolating the tangent function in the equation: given \(3 \tan 3x = \sqrt{3}\), divide both sides by 3 to get \(\tan 3x = \frac{\sqrt{3}}{3}\).

Recall the exact values of tangent for common angles. Since \(\tan \theta = \frac{\sqrt{3}}{3}\), identify the reference angle \(\alpha\) such that \(\tan \alpha = \frac{\sqrt{3}}{3}\). This corresponds to \(\alpha = \frac{\pi}{6}\) radians or 30°.

Write the general solution for \(3x\) using the periodicity of the tangent function, which has period \(\pi\). The solutions are given by \(3x = \alpha + k\pi\), where \(k\) is any integer.

Express \(x\) explicitly by dividing both sides by 3: \(x = \frac{\alpha}{3} + \frac{k\pi}{3}\).

Find all values of \(x\) within the interval \([0, 2\pi)\) by substituting integer values of \(k\) such that \(x\) remains in the interval. List these exact solutions.

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주요 개념

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Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all angle values within a given interval that satisfy the equation. This often requires using inverse trigonometric functions and considering the periodic nature of trig functions to find multiple solutions.

Properties of the Tangent Function

The tangent function, tan(θ), has a period of π, meaning its values repeat every π radians. It is undefined at odd multiples of π/2 and can take any real value. Understanding its periodicity is essential for finding all solutions within a specified interval.

Interval Notation and Angle Measurement

The problem specifies solutions over intervals [0, 2π) for radians and [0°, 360°) for degrees. Knowing how to convert between degrees and radians and interpreting these intervals correctly ensures that all valid solutions are found and expressed within the required domain.

Solve each equation in x over the interval [0, 2π) and each equation in θ over the interval [0°, 360°). Give exact solutions.3 tan 3x = √3