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

Verified step by step guidance

1. Recall the definition of cotangent in terms of sine and cosine: \(\cot x = \frac{\cos x}{\sin x}\).
2. Since the problem asks to write \(\cot x\) in terms of \(\sin x\) only, we need to express \(\cos x\) in terms of \(\sin x\).
3. Use the Pythagorean identity: \(\sin^2 x + \cos^2 x = 1\), which can be rearranged to \(\cos x = \pm \sqrt{1 - \sin^2 x}\).
4. Substitute \(\cos x\) back into the cotangent expression: \(\cot x = \frac{\pm \sqrt{1 - \sin^2 x}}{\sin x}\).
5. Note that the sign depends on the quadrant where \(x\) lies, so the final expression for \(\cot x\) in terms of \(\sin x\) is \(\cot x = \pm \frac{\sqrt{1 - \sin^2 x}}{\sin x}\).

Key Concepts

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.