/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Simplify each expression. \(\si... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 43

Simplify each expression. \(\sin (-x) \cot (-x)\)

Short Answer

Expert verified
The simplified expression is \( \cos(x) \).

Step by step solution

01

Use the sine and cotangent negative angle identities

First, apply the identities for the sine and cotangent of a negative angle: \ \ \ \(\sin(-x) = -\sin(x)\) \ \(\cot(-x) = -\cot(x)\).
02

Substitute the identities into the expression

Next, substitute the identities into the expression: \ \ \(\sin(-x)\cot(-x) = (-\sin(x))(-\cot(x))\).
03

Simplify the expression

Multiply the terms together: \ \ \((-\sin(x))(-\cot(x)) = \sin(x)\cot(x)\).
04

Interpret the cotangent function

Recall that \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). So, we can rewrite \( \sin(x)\cot(x)\) as: \ \ \( \sin(x) \frac{\cos(x)}{\sin(x)} \).
05

Cancel the common terms

Cancel \( \sin(x) \) in the numerator and denominator: \ \ \( \frac{ \sin(x) \cos(x) }{ \sin(x) } = \cos(x) \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sine and Cotangent Identities
Understanding the identities of trigonometric functions like sine and cotangent is crucial for simplifying expressions. Sine, represented as \( \sin(x) \), and cotangent, represented as \( \cot(x) \), have specific properties when dealing with negative angles. Specifically, \( \sin(-x) = -\sin(x) \) and \( \cot(-x) = -\cot(x) \). These properties indicate that both functions are odd functions, meaning they exhibit symmetry around the origin. This is a key piece in understanding how to work with trigonometric expressions involving negative angles.
Negative Angle Identities
Let鈥檚 dive into negative angle identities to see how they simplify our expressions. When you encounter a negative angle in trigonometry, there are certain identities you can use to transform them into more manageable forms. For instance:
  • \( \sin(-x) = -\sin(x) \)
  • \( \cot(-x) = -\cot(x) \)
Applying these identities helps to eliminate the negative sign from the angle, making it easier to perform further simplifications.
Expression Simplification
Once we've applied the appropriate identities, the next step is simplifying the trigonometric expression. For example, in the expression \( \sin(-x) \cot(-x) \), we first apply the identities to get \( (-\sin(x))(-\cot(x)) \).

Then, by multiplying these terms, we get \( \sin(x) \cot(x) \). From the cotangent identity, \( \cot(x) = \frac{\cos(x)}{\sin(x)} \), we can transform the expression into \( \sin(x) \cdot \frac{\cos(x)}{\sin(x)} \).

Finally, we cancel out the common terms - the \( \sin(x) \) in the numerator and denominator, resulting in \( \cos(x) \). Simplification makes the initially complex expression much more manageable!

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Most popular questions from this chapter

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