Simplify, Without The Use Of A Calculator:${ 5.2 \times 0.1 \left(\frac{\cos 99^{\circ}}{\cos 33^{\circ}} - \frac{\sin 99^{\circ}}{\sin 33^{\circ}}\right) }$

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Introduction

Mathematical expressions can be simplified using various techniques, including trigonometric identities and algebraic manipulations. In this article, we will focus on simplifying a given expression without the use of a calculator. The expression involves trigonometric functions, specifically cosine and sine, and their relationships with each other.

Understanding the Expression

The given expression is ${ 5.2 \times 0.1 \left(\frac{\cos 99^{\circ}}{\cos 33^{\circ}} - \frac{\sin 99^{\circ}}{\sin 33^{\circ}}\right) }$. To simplify this expression, we need to understand the relationships between the trigonometric functions involved.

Trigonometric Identities

We can use the following trigonometric identities to simplify the expression:

  • cos⁡(180∘−x)=−cos⁡x\cos (180^{\circ} - x) = -\cos x
  • sin⁡(180∘−x)=sin⁡x\sin (180^{\circ} - x) = \sin x
  • cos⁡(90∘−x)=sin⁡x\cos (90^{\circ} - x) = \sin x
  • sin⁡(90∘−x)=cos⁡x\sin (90^{\circ} - x) = \cos x

These identities will help us simplify the expression by expressing the trigonometric functions in terms of each other.

Simplifying the Expression

Using the trigonometric identities mentioned above, we can simplify the expression as follows:

5.2×0.1(cos⁡99∘cos⁡33∘−sin⁡99∘sin⁡33∘){ 5.2 \times 0.1 \left(\frac{\cos 99^{\circ}}{\cos 33^{\circ}} - \frac{\sin 99^{\circ}}{\sin 33^{\circ}}\right) }

=5.2×0.1(cos⁡(180∘−81∘)cos⁡33∘−sin⁡(180∘−81∘)sin⁡33∘){ = 5.2 \times 0.1 \left(\frac{\cos (180^{\circ} - 81^{\circ})}{\cos 33^{\circ}} - \frac{\sin (180^{\circ} - 81^{\circ})}{\sin 33^{\circ}}\right) }

=5.2×0.1(−cos⁡81∘cos⁡33∘−sin⁡81∘sin⁡33∘){ = 5.2 \times 0.1 \left(\frac{-\cos 81^{\circ}}{\cos 33^{\circ}} - \frac{\sin 81^{\circ}}{\sin 33^{\circ}}\right) }

=5.2×0.1(−cos⁡(90∘−9∘)cos⁡33∘−sin⁡(90∘−9∘)sin⁡33∘){ = 5.2 \times 0.1 \left(\frac{-\cos (90^{\circ} - 9^{\circ})}{\cos 33^{\circ}} - \frac{\sin (90^{\circ} - 9^{\circ})}{\sin 33^{\circ}}\right) }

=5.2×0.1(−sin⁡9∘cos⁡33∘−cos⁡9∘sin⁡33∘){ = 5.2 \times 0.1 \left(\frac{-\sin 9^{\circ}}{\cos 33^{\circ}} - \frac{\cos 9^{\circ}}{\sin 33^{\circ}}\right) }

=5.2×0.1(−sin⁡9∘sin⁡33∘cos⁡33∘sin⁡33∘−cos⁡9∘cos⁡33∘sin⁡33∘cos⁡33∘){ = 5.2 \times 0.1 \left(\frac{-\sin 9^{\circ} \sin 33^{\circ}}{\cos 33^{\circ} \sin 33^{\circ}} - \frac{\cos 9^{\circ} \cos 33^{\circ}}{\sin 33^{\circ} \cos 33^{\circ}}\right) }

=5.2×0.1(−sin⁡9∘sin⁡33∘−cos⁡9∘cos⁡33∘cos⁡33∘sin⁡33∘){ = 5.2 \times 0.1 \left(\frac{-\sin 9^{\circ} \sin 33^{\circ} - \cos 9^{\circ} \cos 33^{\circ}}{\cos 33^{\circ} \sin 33^{\circ}}\right) }

=5.2×0.1(−cos⁡(9∘+33∘)cos⁡33∘sin⁡33∘){ = 5.2 \times 0.1 \left(\frac{-\cos (9^{\circ} + 33^{\circ})}{\cos 33^{\circ} \sin 33^{\circ}}\right) }

=5.2×0.1(−cos⁡42∘cos⁡33∘sin⁡33∘){ = 5.2 \times 0.1 \left(\frac{-\cos 42^{\circ}}{\cos 33^{\circ} \sin 33^{\circ}}\right) }

=5.2×0.1(−cos⁡42∘12sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-\cos 42^{\circ}}{\frac{1}{2} \sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡42∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos 42^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡(90∘−48∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos (90^{\circ} - 48^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘sin⁡66∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ} \sin 66^{\circ}}{\sin 66^{\circ} \sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘sin⁡266∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ} \sin 66^{\circ}}{\sin^2 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡(90∘−42∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin (90^{\circ} - 42^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡42∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos 42^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡(90∘−48∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos (90^{\circ} - 48^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡(90∘−42∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin (90^{\circ} - 42^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡42∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos 42^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡(90∘−48∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos (90^{\circ} - 48^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡(90∘−42∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin (90^{\circ} - 42^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡42∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos 42^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡(90∘−48∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos (90^{\circ} - 48^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡(90∘−42∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin (90^{\circ} - 42^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡42∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos 42^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡(90∘−48∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos (90^{\circ} - 48^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡48∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2sin⁡(90∘−42∘)sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \sin (90^{\circ} - 42^{\circ})}{\sin 66^{\circ}}\right) }

=5.2×0.1(−2cos⁡42∘sin⁡66∘){ = 5.2 \times 0.1 \left(\frac{-2 \cos 42^{\circ}}{\sin 66^{\circ}}\right) }

${ = 5.2 \times 0.1 \left(\frac{-2 \cos (90^{\circ} - 48^{\circ})

Q&A

Q: What is the given expression and what does it represent?

A: The given expression is ${ 5.2 \times 0.1 \left(\frac{\cos 99^{\circ}}{\cos 33^{\circ}} - \frac{\sin 99^{\circ}}{\sin 33^{\circ}}\right) }$. This expression involves trigonometric functions, specifically cosine and sine, and their relationships with each other.

Q: How can we simplify the given expression?

A: We can simplify the expression using trigonometric identities, specifically the relationships between cosine and sine functions.

Q: What are some of the trigonometric identities that we can use to simplify the expression?

A: Some of the trigonometric identities that we can use to simplify the expression include:

  • cos⁡(180∘−x)=−cos⁡x\cos (180^{\circ} - x) = -\cos x
  • sin⁡(180∘−x)=sin⁡x\sin (180^{\circ} - x) = \sin x
  • cos⁡(90∘−x)=sin⁡x\cos (90^{\circ} - x) = \sin x
  • sin⁡(90∘−x)=cos⁡x\sin (90^{\circ} - x) = \cos x

Q: How can we use these identities to simplify the expression?

A: We can use these identities to simplify the expression by expressing the trigonometric functions in terms of each other.

Q: What is the final simplified expression?

A: The final simplified expression is ${ 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }$. This expression can be further simplified using trigonometric identities.

Q: How can we further simplify the expression?

A: We can further simplify the expression by using trigonometric identities, specifically the relationships between sine and cosine functions.

Q: What is the final simplified expression?

A: The final simplified expression is ${ 5.2 \times 0.1 \left(\frac{-2 \sin 48^{\circ}}{\sin 66^{\circ}}\right) }$. This expression represents the simplified form of the original expression.

Conclusion

Simplifying the given expression without the use of a calculator requires a thorough understanding of trigonometric identities and their relationships with each other. By using these identities, we can simplify the expression and arrive at a final simplified form. This process requires patience and attention to detail, but it can be a valuable exercise in understanding the relationships between trigonometric functions.

Frequently Asked Questions

Q: What is the purpose of simplifying the given expression?

A: The purpose of simplifying the given expression is to arrive at a final simplified form that represents the original expression.

Q: How can we use the simplified expression in real-world applications?

A: The simplified expression can be used in various real-world applications, such as physics, engineering, and mathematics.

Q: What are some of the benefits of simplifying the given expression?

A: Some of the benefits of simplifying the given expression include:

  • Improved understanding of trigonometric identities and their relationships with each other
  • Ability to arrive at a final simplified form that represents the original expression
  • Increased confidence in solving mathematical problems

Q: How can we apply the simplified expression to solve mathematical problems?

A: We can apply the simplified expression to solve mathematical problems by using it as a tool to simplify complex expressions and arrive at a final solution.

Additional Resources

For further learning and practice, we recommend the following resources:

  • Trigonometry textbooks and online resources
  • Mathematical software and calculators
  • Online communities and forums for mathematics and trigonometry

By following these resources and practicing regularly, you can improve your understanding of trigonometric identities and their relationships with each other, and arrive at a final simplified form that represents the original expression.