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Chapter 3: Problem 59

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

Short Answer

Expert verified
The expression simplifies to \(\

Step by step solution

01

- Understand the properties of trigonometric functions

Recall that \(\sin(-x) = -\sin(x)\) and \(\cot(-x) = -\cot(x)\). These are standard identities for sine and cotangent functions considering the odd property of these functions.
02

- Apply the identities

Substitute the identities for \(\sin(-x)\) and \(\cot(-x)\): \(\sin(-x) \cot(-x) = (-\sin(x))(-\cot(x))\).
03

- Simplify the expression

Simplify the expression by multiplying the two negatives: \(\begin{align*} \sin(-x) \cot(-x) &= (-\sin(x))(-\cot(x)) \ &= \sin(x) \cot(x). \end{align*}\).

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Key Concepts

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

The Sine Function
The sine function, commonly written as \(\sin(x)\), plays a vital role in trigonometry. It measures the y-coordinate of a point on the unit circle as the angle x is varied. Key properties of the sine function include:
  • Periodicity: The sine function has a period of \(\2\pi\). This means that \(\sin(x + 2\pi) = \sin(x)\).
  • Range: The values of the sine function are always between -1 and 1, inclusive.
  • Odd Function: One important property is that the sine function is odd, which means \(\sin(-x) = -\sin(x)\).
Understanding these properties helps in simplifying expressions involving sine, like in the given problem.
The Cotangent Function
The cotangent function, represented as \(\cot(x)\), is the reciprocal of the tangent function. Mathematically, it鈥檚 defined as \(\cot(x) = \frac{1}{\tan(x)}\), or alternatively, \(\cot(x) = \frac{\cos(x)}{\sin(x)}\). Important properties include:
  • Periodicity: \(\cot(x)\) has a period of \(\pi\), meaning \(\cot(x + \pi) = \cot(x)\).
  • Range: Unlike sine, the cotangent function can take any real value.
  • Odd Function: The cotangent is also an odd function, satisfying \(\cot(-x) = -\cot(x)\).
Similar to sine, these properties assist in transforming and simplifying trigonometric expressions, like replacing \(\cot(-x)\) with \(\-\cot(x)\) as seen in the problem.
Odd Functions in Trigonometry
In trigonometry, a function \(\f(x)\) is odd if it satisfies the condition \(\f(-x) = -\f(x)\). This property simplifies many trigonometric expressions, particularly for functions like sine and cotangent. Key examples of odd trigonometric functions include:
  • \(\sin(x)\): Since \(\sin(-x) = -\sin(x)\), the sine function is odd.
  • \(\cot(x)\): Similarly, since \(\cot(-x) = -\cot(x)\), the cotangent function is also odd.
When simplifying expressions involving these functions, recognizing their odd nature can lead to quicker solutions. For instance, in the given exercise, using odd function properties allowed us to replace \(\sin(-x)\) and \(\cot(-x)\) efficiently and simplify the expression correctly.

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Most popular questions from this chapter

Write each expression as a function of \(\alpha\) alone. $$ \cos (\alpha-\pi / 2) $$

Prove that each equation is an identity: $$ \sin 2 A \sin 2 B=\sin ^{2}(A+B)-\sin ^{2}(A-B) $$

Find the degree measure of the acute angle between the lines \(y=\frac{2}{3} x\) and \(y=5 x-13\)

Prove that each equation is an identity. \((\sin \alpha-\cos \alpha)^{2}=1-\sin 2 \alpha\)

The equation \(f_{1}(x)=f_{2}(x)\) is an identity if and only if the graphs of \(y=f_{1}(x)\) and \(y=f_{2}(x)\) coincide at all values of \(x\) for which both sides are defined. Graph \(y=f_{1}(x)\) and \(y=f_{2}(x)\) on the same screen of your calculator for each of the following equations. From the graphs, make a conjecture as to whether each equation is an identity, then prove your conjecture. $$ \frac{\cos (-x)}{1-\sin x}=\frac{1-\sin (-x)}{\cos x} $$
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