Find the exact values of s in the given interval that satisfy the given condition.

[-2π , π) ; 3 tan² s = 1

검증된 단계별 안내

Start by writing down the given equation: \(3 \tan^{2} s = 1\).

Isolate \(\tan^{2} s\) by dividing both sides of the equation by 3: \(\tan^{2} s = \frac{1}{3}\).

Take the square root of both sides to solve for \(\tan s\): \(\tan s = \pm \frac{1}{\sqrt{3}}\).

Recall that \(\tan s = \pm \frac{1}{\sqrt{3}}\) corresponds to specific reference angles where tangent has these values. Identify these angles within one full period of tangent, which is \(\pi\).

Find all values of \(s\) in the interval \([-2\pi, \pi)\) by adding integer multiples of \(\pi\) to the reference angles, and then select those that lie within the given interval.

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주요 개념

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

Solving Trigonometric Equations

Solving trigonometric equations involves finding all angle values that satisfy the given equation within a specified interval. This often requires isolating the trigonometric function and using inverse functions or known identities to determine possible solutions.

Properties of the Tangent Function

The tangent function, tan(s), is periodic with period π and is undefined at odd multiples of π/2. Understanding its behavior, including its range and periodicity, is essential for finding all solutions within a given interval.

Interval Notation and Solution Sets

The interval [-2π, π) specifies the domain in which solutions must be found, including -2π but excluding π. Correctly interpreting this interval ensures that all valid solutions are identified and extraneous ones outside the range are excluded.

Find the exact values of s in the given interval that satisfy the given condition.[-2π , π) ; 3 tan² s = 1