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

검증된 단계별 안내

Recall the definition of cotangent in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\).

Since the problem asks to write \(\cot x\) in terms of \(\sin x\) only, we need to express \(\cos x\) in terms of \(\sin x\).

Use the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\), which can be rearranged to \(\cos x = \pm \sqrt{1 - \sin^2 x}\).

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

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

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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주요 개념

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

Definition of Cotangent

Cotangent (cot x) is the reciprocal of the tangent function. It can be expressed as cot x = cos x / sin x, which relates cotangent directly to sine and cosine functions.

Relationship Between Sine and Cosine

Sine and cosine are fundamental trigonometric functions related by the Pythagorean identity: sin²x + cos²x = 1. This identity allows expressing one function in terms of the other, which is useful when rewriting cotangent solely in terms of sine.

Algebraic Manipulation of Trigonometric Expressions

Transforming trigonometric expressions often requires algebraic skills such as substitution and rearrangement. To write cot x in terms of sin x, one must manipulate the expression cot x = cos x / sin x using identities and algebraic steps.

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