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. tan x sec x = 2 tan x

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Start with the given equation: \(\tan x \sec x = 2 \tan x\).

Bring all terms to one side to set the equation to zero: \(\tan x \sec x - 2 \tan x = 0\).

Factor out the common factor \(\tan x\): \(\tan x (\sec x - 2) = 0\).

Set each factor equal to zero and solve separately: 1) \(\tan x = 0\) 2) \(\sec x - 2 = 0\).

For \(\tan x = 0\), find all \(x\) in \([0, 2\pi)\) where tangent is zero. For \(\sec x - 2 = 0\), rewrite as \(\sec x = 2\), then use the identity \(\sec x = \frac{1}{\cos x}\) to find \(\cos x = \frac{1}{2}\) and solve for \(x\) in \([0, 2\pi)\).

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주요 개념

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

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values within their domains. In this problem, recognizing identities like sec x = 1/cos x and the relationship between tan x and sin x/cos x helps simplify and solve the equation efficiently.

Solving Trigonometric Equations

Solving trigonometric equations involves isolating the trigonometric function and finding all solutions within a given interval. This often requires factoring, using identities, and considering the domain restrictions to find exact or approximate values of x.

Interval and Solution Restrictions

When solving trigonometric equations on a specific interval like [0, 2π), it is essential to find all solutions within that range. Additionally, one must consider where functions are undefined (e.g., sec x undefined when cos x = 0) to exclude invalid solutions.