Use substitution to determine whether the given x-value is a solution of the equation.
$\tan 2 x = - \frac{\sqrt{3}}{3},$

검증된 단계별 안내

First, substitute the given value of $x = \frac{5\pi}{12}$ into the left side of the equation $\tan 2x$. This means calculating $\tan\left(2 \times \frac{5\pi}{12}\right)$.

Simplify the expression inside the tangent function: $2 \times \frac{5\pi}{12} = \frac{10\pi}{12} = \frac{5\pi}{6}$, so you need to find $\tan\left(\frac{5\pi}{6}\right)$.

Recall or use the unit circle to find the exact value of $\tan\left(\frac{5\pi}{6}\right)$. Remember that $\frac{5\pi}{6}$ is in the second quadrant where tangent is negative.

Next, evaluate the right side of the equation, which is $-\frac{\sqrt{3}}{3}$. This is a constant value you can compare with the left side.

Finally, compare the value of $\tan\left(\frac{5\pi}{6}\right)$ with $-\frac{\sqrt{3}}{3}$. If they are equal, then $x = \frac{5\pi}{12}$ is a solution; if not, it is not a solution.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Tangent Function and Its Properties

The tangent function, tan(θ), is defined as the ratio of sine to cosine (sin θ / cos θ). It is periodic with period π, meaning tan(θ + π) = tan(θ). Understanding how to evaluate tangent at specific angles, especially multiples of π, is essential for solving equations involving tan(2x).

Substitution Method in Trigonometric Equations

Substitution involves replacing the variable x with a given value to verify if it satisfies the equation. Here, substituting x = 5π/12 into tan(2x) allows direct evaluation to check if it equals the given expression. This method helps confirm whether the proposed x-value is a solution.

Simplifying and Evaluating Trigonometric Expressions

Evaluating tan(2x) at x = 5π/12 requires simplifying the angle 2x = 5π/6 and then calculating tan(5π/6). Knowing exact values of tangent at standard angles and simplifying radicals like √3/3 is crucial to compare both sides of the equation accurately.

Use substitution to determine whether the given x-value is a solution of the equation. tan2x=−33,x=5π12\(\tan\) 2x = -\(\frac{\sqrt{3}\)}{3}, \(\quad\) x = \(\frac{5\pi}{12}\)