Which Expression Is Equivalent To Sin ⁡ 2 X − Sin ⁡ X − 2 Sin ⁡ X − 2 \frac{\sin ^2 X-\sin X-2}{\sin X-2} S I N X − 2 S I N 2 X − S I N X − 2 ?A. Sin ⁡ X + 1 \sin X+1 Sin X + 1 B. Sin ⁡ X − 1 \sin X-1 Sin X − 1 C. Sin ⁡ 2 X \sin ^2 X Sin 2 X D. − Sin ⁡ 2 X -\sin ^2 X − Sin 2 X

Mar 10, 2025 by ADMIN 280 views

Simplifying Trigonometric Expressions: A Step-by-Step Guide

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Introduction

Trigonometric expressions are a fundamental part of mathematics, and simplifying them is a crucial skill for students and professionals alike. In this article, we will explore the process of simplifying a given trigonometric expression, and we will use this knowledge to determine which of the provided options is equivalent to the given expression.

The Given Expression

The given expression is $\frac{\sin ^2 x-\sin x-2}{\sin x-2}$ . Our goal is to simplify this expression and determine which of the provided options is equivalent to it.

Step 1: Factor the Numerator

To simplify the given expression, we can start by factoring the numerator. We can use the quadratic formula or factorization techniques to factor the numerator.

import sympy as sp
x = sp.symbols('x')

numerator = sp.sin(x)**2 - sp.sin(x) - 2

factored_numerator = sp.factor(numerator)
print(factored_numerator)

The factored numerator is $(\sin x + 1)(\sin x - 2)$ .

Step 2: Cancel Common Factors

Now that we have factored the numerator, we can cancel common factors between the numerator and the denominator. In this case, we can cancel the factor of $(\sin x - 2)$ .

# Cancel common factors
simplified_expression = factored_numerator / (sp.sin(x) - 2)
print(simplified_expression)

The simplified expression is $\sin x + 1$ .

Conclusion

Based on our step-by-step simplification process, we have determined that the given expression $\frac{\sin ^2 x-\sin x-2}{\sin x-2}$ is equivalent to $\sin x + 1$ .

Answer

The correct answer is A. $\sin x + 1$ .

Discussion

In this article, we have demonstrated the process of simplifying a trigonometric expression using factoring and canceling common factors. We have also used this knowledge to determine which of the provided options is equivalent to the given expression.

Tips and Tricks

When simplifying trigonometric expressions, it is often helpful to factor the numerator and cancel common factors between the numerator and the denominator.
Use the quadratic formula or factorization techniques to factor the numerator.
Cancel common factors between the numerator and the denominator to simplify the expression.

Final Thoughts

Simplifying trigonometric expressions is a crucial skill for students and professionals alike. By following the step-by-step process outlined in this article, you can simplify even the most complex trigonometric expressions and determine which of the provided options is equivalent to the given expression.

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Introduction

In our previous article, we explored the process of simplifying a given trigonometric expression. We used factoring and canceling common factors to simplify the expression and determine which of the provided options is equivalent to it. In this article, we will answer some frequently asked questions about trigonometric expression simplification.

Q&A

Q: What is the first step in simplifying a trigonometric expression?

A: The first step in simplifying a trigonometric expression is to factor the numerator. This can be done using the quadratic formula or factorization techniques.

Q: How do I factor the numerator of a trigonometric expression?

A: To factor the numerator of a trigonometric expression, you can use the quadratic formula or factorization techniques. For example, if the numerator is $\sin^2 x - \sin x - 2$ , you can factor it as $(\sin x + 1)(\sin x - 2)$ .

Q: What is the next step after factoring the numerator?

A: After factoring the numerator, the next step is to cancel common factors between the numerator and the denominator. This can help simplify the expression and make it easier to work with.

Q: How do I cancel common factors between the numerator and the denominator?

A: To cancel common factors between the numerator and the denominator, you can simply divide both the numerator and the denominator by the common factor. For example, if the numerator is $(\sin x + 1)(\sin x - 2)$ and the denominator is $(\sin x - 2)$ , you can cancel the common factor of $(\sin x - 2)$ to get $\sin x + 1$ .

Q: What are some common mistakes to avoid when simplifying trigonometric expressions?

A: Some common mistakes to avoid when simplifying trigonometric expressions include:

Not factoring the numerator
Not canceling common factors between the numerator and the denominator
Making algebraic errors when simplifying the expression

Q: How can I practice simplifying trigonometric expressions?

A: You can practice simplifying trigonometric expressions by working through examples and exercises. You can also try simplifying expressions on your own and then checking your work to see if you made any mistakes.

Q: What are some real-world applications of trigonometric expression simplification?

A: Trigonometric expression simplification has many real-world applications, including:

Calculating distances and angles in navigation and surveying
Modeling population growth and decay in biology and economics
Analyzing data in statistics and data analysis

Conclusion

In this article, we have answered some frequently asked questions about trigonometric expression simplification. We have covered topics such as factoring the numerator, canceling common factors, and avoiding common mistakes. We have also discussed some real-world applications of trigonometric expression simplification.

Tips and Tricks

Practice simplifying trigonometric expressions regularly to build your skills and confidence.
Use online resources and tools, such as calculators and software, to help you simplify expressions and check your work.
Don't be afraid to ask for help if you get stuck or make a mistake.