In Exercises 55–58, use a graph to solve each equation for -2π ≤ x ≤ 2π. csc x = 1

검증된 단계별 안내

Recall that the cosecant function is the reciprocal of the sine function, so \(\csc x = \frac{1}{\sin x}\). Therefore, the equation \(\csc x = 1\) can be rewritten as \(\frac{1}{\sin x} = 1\).

From the equation \(\frac{1}{\sin x} = 1\), multiply both sides by \(\sin x\) (noting \(\sin x \neq 0\)) to get \(1 = \sin x\).

Now, solve the equation \(\sin x = 1\) for \(x\) in the interval \(-2\pi \leq x \leq 2\pi\) by identifying where the sine function reaches the value 1 on its graph.

Recall that \(\sin x = 1\) at \(x = \frac{\pi}{2} + 2k\pi\) for any integer \(k\). Find all such \(x\) values within the given interval by substituting integer values for \(k\).

List all solutions found in the interval \(-2\pi \leq x \leq 2\pi\) as the final answer to the equation \(\csc x = 1\).

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

주요 개념

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

Understanding the Cosecant Function

The cosecant function, csc x, is the reciprocal of the sine function, defined as csc x = 1/sin x. It is undefined where sin x = 0, and its values correspond to the reciprocal of sine values. Recognizing this relationship helps in solving equations involving csc x.

Graphing Trigonometric Functions

Graphing csc x involves plotting the reciprocal of the sine curve, which has vertical asymptotes where sine is zero. Understanding the shape and key points of the csc x graph allows one to visually identify solutions to equations like csc x = 1 within a given interval.

Solving Trigonometric Equations on a Given Interval

Solving csc x = 1 over -2π ≤ x ≤ 2π requires finding all x-values where csc x equals 1 within this domain. This involves identifying corresponding sine values (sin x = 1) and considering the periodicity of the sine and cosecant functions to list all valid solutions.