Simplify The Expression:$\[ \frac{\sin(180 + \theta) \cdot \cos(360 - \theta) \cdot \tan(6 - \theta)}{2 \cdot \cos(90 + \theta) \cdot \sin(90 + \theta)} \\]

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Introduction

Trigonometric identities are a fundamental concept in mathematics, and they play a crucial role in simplifying complex expressions. In this article, we will focus on simplifying the given expression using various trigonometric identities. The expression involves sine, cosine, and tangent functions, and we will use these identities to simplify it step by step.

Understanding the Expression

The given expression is:

sin⁡(180+θ)⋅cos⁡(360−θ)⋅tan⁡(6−θ)2⋅cos⁡(90+θ)⋅sin⁡(90+θ)\frac{\sin(180 + \theta) \cdot \cos(360 - \theta) \cdot \tan(6 - \theta)}{2 \cdot \cos(90 + \theta) \cdot \sin(90 + \theta)}

This expression involves various trigonometric functions, including sine, cosine, and tangent. We will use these functions to simplify the expression.

Simplifying the Expression

To simplify the expression, we will use various trigonometric identities. We will start by simplifying the numerator and denominator separately.

Simplifying the Numerator

The numerator of the expression is:

sin⁡(180+θ)⋅cos⁡(360−θ)⋅tan⁡(6−θ)\sin(180 + \theta) \cdot \cos(360 - \theta) \cdot \tan(6 - \theta)

We can simplify this expression using the following trigonometric identities:

  • sin⁡(180+θ)=−sin⁡(θ)\sin(180 + \theta) = -\sin(\theta)
  • cos⁡(360−θ)=cos⁡(θ)\cos(360 - \theta) = \cos(\theta)
  • tan⁡(6−θ)=sin⁡(6−θ)cos⁡(6−θ)\tan(6 - \theta) = \frac{\sin(6 - \theta)}{\cos(6 - \theta)}

Using these identities, we can simplify the numerator as follows:

sin⁡(180+θ)⋅cos⁡(360−θ)⋅tan⁡(6−θ)\sin(180 + \theta) \cdot \cos(360 - \theta) \cdot \tan(6 - \theta)

=−sin⁡(θ)⋅cos⁡(θ)⋅sin⁡(6−θ)cos⁡(6−θ)= -\sin(\theta) \cdot \cos(\theta) \cdot \frac{\sin(6 - \theta)}{\cos(6 - \theta)}

=−sin⁡(θ)⋅cos⁡(θ)⋅sin⁡(6−θ)cos⁡(6−θ)= -\frac{\sin(\theta) \cdot \cos(\theta) \cdot \sin(6 - \theta)}{\cos(6 - \theta)}

Simplifying the Denominator

The denominator of the expression is:

2⋅cos⁡(90+θ)⋅sin⁡(90+θ)2 \cdot \cos(90 + \theta) \cdot \sin(90 + \theta)

We can simplify this expression using the following trigonometric identities:

  • cos⁡(90+θ)=−sin⁡(θ)\cos(90 + \theta) = -\sin(\theta)
  • sin⁡(90+θ)=cos⁡(θ)\sin(90 + \theta) = \cos(\theta)

Using these identities, we can simplify the denominator as follows:

2⋅cos⁡(90+θ)⋅sin⁡(90+θ)2 \cdot \cos(90 + \theta) \cdot \sin(90 + \theta)

=2⋅(−sin⁡(θ))⋅cos⁡(θ)= 2 \cdot (-\sin(\theta)) \cdot \cos(\theta)

=−2⋅sin⁡(θ)⋅cos⁡(θ)= -2 \cdot \sin(\theta) \cdot \cos(\theta)

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to simplify the expression.

sin⁡(180+θ)⋅cos⁡(360−θ)⋅tan⁡(6−θ)2⋅cos⁡(90+θ)⋅sin⁡(90+θ)\frac{\sin(180 + \theta) \cdot \cos(360 - \theta) \cdot \tan(6 - \theta)}{2 \cdot \cos(90 + \theta) \cdot \sin(90 + \theta)}

=−sin⁡(θ)⋅cos⁡(θ)⋅sin⁡(6−θ)cos⁡(6−θ)−2⋅sin⁡(θ)⋅cos⁡(θ)= \frac{-\frac{\sin(\theta) \cdot \cos(\theta) \cdot \sin(6 - \theta)}{\cos(6 - \theta)}}{-2 \cdot \sin(\theta) \cdot \cos(\theta)}

Canceling Out Common Factors

We can simplify the expression further by canceling out common factors.

−sin⁡(θ)⋅cos⁡(θ)⋅sin⁡(6−θ)cos⁡(6−θ)−2⋅sin⁡(θ)⋅cos⁡(θ)\frac{-\frac{\sin(\theta) \cdot \cos(\theta) \cdot \sin(6 - \theta)}{\cos(6 - \theta)}}{-2 \cdot \sin(\theta) \cdot \cos(\theta)}

=sin⁡(6−θ)2⋅cos⁡(6−θ)= \frac{\sin(6 - \theta)}{2 \cdot \cos(6 - \theta)}

Final Simplification

We can simplify the expression further by using the following trigonometric identity:

  • sin⁡(α)cos⁡(α)=tan⁡(α)\frac{\sin(\alpha)}{\cos(\alpha)} = \tan(\alpha)

Using this identity, we can simplify the expression as follows:

sin⁡(6−θ)2⋅cos⁡(6−θ)\frac{\sin(6 - \theta)}{2 \cdot \cos(6 - \theta)}

=12⋅tan⁡(6−θ)= \frac{1}{2} \cdot \tan(6 - \theta)

Conclusion

In this article, we simplified the given expression using various trigonometric identities. We started by simplifying the numerator and denominator separately and then combined them to simplify the expression. We also canceled out common factors and used trigonometric identities to simplify the expression further. The final simplified expression is 12⋅tan⁡(6−θ)\frac{1}{2} \cdot \tan(6 - \theta).

Final Answer

The final answer is 12⋅tan⁡(6−θ)\boxed{\frac{1}{2} \cdot \tan(6 - \theta)}.

Introduction

In our previous article, we simplified the given expression using various trigonometric identities. In this article, we will provide a Q&A section to help readers understand the concept better.

Frequently Asked Questions

Q1: What is the main concept behind simplifying the expression?

A1: The main concept behind simplifying the expression is to use various trigonometric identities to reduce the complexity of the expression.

Q2: What are some common trigonometric identities used in simplifying expressions?

A2: Some common trigonometric identities used in simplifying expressions include:

  • sin⁡(A+B)=sin⁡(A)cos⁡(B)+cos⁡(A)sin⁡(B)\sin(A + B) = \sin(A) \cos(B) + \cos(A) \sin(B)
  • cos⁡(A+B)=cos⁡(A)cos⁡(B)−sin⁡(A)sin⁡(B)\cos(A + B) = \cos(A) \cos(B) - \sin(A) \sin(B)
  • tan⁡(A+B)=tan⁡(A)+tan⁡(B)1−tan⁡(A)tan⁡(B)\tan(A + B) = \frac{\tan(A) + \tan(B)}{1 - \tan(A) \tan(B)}
  • sin⁡(A−B)=sin⁡(A)cos⁡(B)−cos⁡(A)sin⁡(B)\sin(A - B) = \sin(A) \cos(B) - \cos(A) \sin(B)
  • cos⁡(A−B)=cos⁡(A)cos⁡(B)+sin⁡(A)sin⁡(B)\cos(A - B) = \cos(A) \cos(B) + \sin(A) \sin(B)

Q3: How do I simplify a complex expression using trigonometric identities?

A3: To simplify a complex expression using trigonometric identities, follow these steps:

  1. Identify the trigonometric functions involved in the expression.
  2. Use the appropriate trigonometric identities to simplify the expression.
  3. Cancel out common factors and simplify the expression further.

Q4: What is the final simplified expression for the given problem?

A4: The final simplified expression for the given problem is 12⋅tan⁡(6−θ)\frac{1}{2} \cdot \tan(6 - \theta).

Q5: Can I use trigonometric identities to simplify expressions involving other mathematical functions?

A5: Yes, trigonometric identities can be used to simplify expressions involving other mathematical functions, such as exponential and logarithmic functions.

Q6: How do I apply trigonometric identities to simplify expressions involving multiple angles?

A6: To apply trigonometric identities to simplify expressions involving multiple angles, use the following steps:

  1. Identify the trigonometric functions involved in the expression.
  2. Use the appropriate trigonometric identities to simplify the expression.
  3. Simplify the expression further by canceling out common factors.

Q7: What are some common mistakes to avoid when simplifying expressions using trigonometric identities?

A7: Some common mistakes to avoid when simplifying expressions using trigonometric identities include:

  • Not identifying the trigonometric functions involved in the expression.
  • Not using the appropriate trigonometric identities to simplify the expression.
  • Not canceling out common factors and simplifying the expression further.

Conclusion

In this article, we provided a Q&A section to help readers understand the concept of simplifying expressions using trigonometric identities. We covered various topics, including common trigonometric identities, simplifying complex expressions, and applying trigonometric identities to simplify expressions involving multiple angles.

Final Answer

The final answer is 12⋅tan⁡(6−θ)\boxed{\frac{1}{2} \cdot \tan(6 - \theta)}.