Perform each transformation. See Example 2.
Write sec x in terms of sin x.

검증된 단계별 안내

Recall the definition of secant in terms of cosine: \(\sec x = \frac{1}{\cos x}\).

Use the Pythagorean identity relating sine and cosine: \(\sin^2 x + \cos^2 x = 1\).

Solve the identity for \(\cos x\): \(\cos x = \pm \sqrt{1 - \sin^2 x}\).

Substitute \(\cos x\) into the secant expression: \(\sec x = \frac{1}{\pm \sqrt{1 - \sin^2 x}}\).

Note that the sign depends on the quadrant of \(x\), so the expression for \(\sec x\) in terms of \(\sin x\) is \(\sec x = \pm \frac{1}{\sqrt{1 - \sin^2 x}}\).

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주요 개념

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

Reciprocal Trigonometric Functions

The secant function, sec x, is defined as the reciprocal of the cosine function, i.e., sec x = 1/cos x. Understanding this relationship is essential to rewrite sec x in terms of other trigonometric functions.

Pythagorean Identity

The fundamental identity sin²x + cos²x = 1 allows expressing cosine in terms of sine: cos x = ±√(1 - sin²x). This identity is crucial for converting sec x into an expression involving sin x.

Domain and Sign Considerations

When expressing sec x in terms of sin x, the sign of cos x (and thus sec x) depends on the quadrant of x. Recognizing the domain restrictions ensures the correct sign is chosen for the square root in the transformation.

Perform each transformation. See Example 2.Write sec x in terms of sin x.