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

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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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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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