Simplify Without The Use Of A Calculator:${ \frac{\cos 225^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}} }$

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Introduction

In this article, we will explore a mathematical expression involving trigonometric functions and simplify it without the use of a calculator. The expression is given as cos⁑225βˆ˜β‹…sin⁑(βˆ’135∘)βˆ’sin⁑330∘tan⁑225∘\frac{\cos 225^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}}. We will use various trigonometric identities and properties to simplify this expression step by step.

Understanding the Trigonometric Functions

Before we proceed with simplifying the expression, let's recall the definitions of the trigonometric functions involved:

  • cos⁑θ\cos \theta is the cosine of angle ΞΈ\theta, which is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.
  • sin⁑θ\sin \theta is the sine of angle ΞΈ\theta, which is the ratio of the opposite side to the hypotenuse in a right-angled triangle.
  • tan⁑θ\tan \theta is the tangent of angle ΞΈ\theta, which is the ratio of the opposite side to the adjacent side in a right-angled triangle.

Simplifying the Expression

To simplify the expression, we will use the following trigonometric identities:

  • cos⁑(180∘+ΞΈ)=βˆ’cos⁑θ\cos (180^{\circ} + \theta) = -\cos \theta
  • sin⁑(180∘+ΞΈ)=sin⁑θ\sin (180^{\circ} + \theta) = \sin \theta
  • tan⁑(180∘+ΞΈ)=tan⁑θ\tan (180^{\circ} + \theta) = \tan \theta

Using these identities, we can rewrite the expression as:

cos⁑225βˆ˜β‹…sin⁑(βˆ’135∘)βˆ’sin⁑330∘tan⁑225∘\frac{\cos 225^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}}

=cos⁑(180∘+45∘)β‹…sin⁑(βˆ’135∘)βˆ’sin⁑(180∘+150∘)tan⁑(180∘+45∘)= \frac{\cos (180^{\circ} + 45^{\circ}) \cdot \sin \left(-135^{\circ}\right) - \sin (180^{\circ} + 150^{\circ})}{\tan (180^{\circ} + 45^{\circ})}

=βˆ’cos⁑45βˆ˜β‹…sin⁑(βˆ’135∘)βˆ’sin⁑330∘tan⁑225∘= \frac{-\cos 45^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}}

Using Trigonometric Properties

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

  • sin⁑(βˆ’ΞΈ)=βˆ’sin⁑θ\sin (-\theta) = -\sin \theta
  • cos⁑θ⋅sin⁑θ=12sin⁑2ΞΈ\cos \theta \cdot \sin \theta = \frac{1}{2} \sin 2\theta

Using these properties, we can rewrite the expression as:

=βˆ’cos⁑45βˆ˜β‹…βˆ’sin⁑135βˆ˜βˆ’sin⁑330∘tan⁑225∘= \frac{-\cos 45^{\circ} \cdot -\sin 135^{\circ} - \sin 330^{\circ}}{\tan 225^{\circ}}

=cos⁑45βˆ˜β‹…sin⁑135βˆ˜βˆ’sin⁑330∘tan⁑225∘= \frac{\cos 45^{\circ} \cdot \sin 135^{\circ} - \sin 330^{\circ}}{\tan 225^{\circ}}

Simplifying the Expression Further

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

  • cos⁑θ⋅sin⁑θ=12sin⁑2ΞΈ\cos \theta \cdot \sin \theta = \frac{1}{2} \sin 2\theta
  • sin⁑(180∘+ΞΈ)=sin⁑θ\sin (180^{\circ} + \theta) = \sin \theta

Using these identities, we can rewrite the expression as:

=12sin⁑90βˆ˜βˆ’sin⁑330∘tan⁑225∘= \frac{\frac{1}{2} \sin 90^{\circ} - \sin 330^{\circ}}{\tan 225^{\circ}}

=12β‹…1βˆ’sin⁑330∘tan⁑225∘= \frac{\frac{1}{2} \cdot 1 - \sin 330^{\circ}}{\tan 225^{\circ}}

Evaluating the Expression

We can evaluate the expression by using the following trigonometric values:

  • sin⁑90∘=1\sin 90^{\circ} = 1
  • sin⁑330∘=βˆ’12\sin 330^{\circ} = -\frac{1}{2}

Using these values, we can rewrite the expression as:

=12βˆ’(βˆ’12)tan⁑225∘= \frac{\frac{1}{2} - \left(-\frac{1}{2}\right)}{\tan 225^{\circ}}

=12+12tan⁑225∘= \frac{\frac{1}{2} + \frac{1}{2}}{\tan 225^{\circ}}

=1tan⁑225∘= \frac{1}{\tan 225^{\circ}}

Final Simplification

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

  • tan⁑θ=sin⁑θcos⁑θ\tan \theta = \frac{\sin \theta}{\cos \theta}

Using this identity, we can rewrite the expression as:

=1sin⁑225∘cos⁑225∘= \frac{1}{\frac{\sin 225^{\circ}}{\cos 225^{\circ}}}

=cos⁑225∘sin⁑225∘= \frac{\cos 225^{\circ}}{\sin 225^{\circ}}

Conclusion

In this article, we simplified the expression cos⁑225βˆ˜β‹…sin⁑(βˆ’135∘)βˆ’sin⁑330∘tan⁑225∘\frac{\cos 225^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}} without the use of a calculator. We used various trigonometric identities and properties to simplify the expression step by step. The final simplified expression is cos⁑225∘sin⁑225∘\frac{\cos 225^{\circ}}{\sin 225^{\circ}}.

Introduction

In our previous article, we simplified the expression cos⁑225βˆ˜β‹…sin⁑(βˆ’135∘)βˆ’sin⁑330∘tan⁑225∘\frac{\cos 225^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}} without the use of a calculator. We used various trigonometric identities and properties to simplify the expression step by step. In this article, we will answer some frequently asked questions related to the simplification of this expression.

Q&A

Q: What is the final simplified expression of cos⁑225βˆ˜β‹…sin⁑(βˆ’135∘)βˆ’sin⁑330∘tan⁑225∘\frac{\cos 225^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}}?

A: The final simplified expression is cos⁑225∘sin⁑225∘\frac{\cos 225^{\circ}}{\sin 225^{\circ}}.

Q: How did you simplify the expression without using a calculator?

A: We used various trigonometric identities and properties, such as cos⁑(180∘+ΞΈ)=βˆ’cos⁑θ\cos (180^{\circ} + \theta) = -\cos \theta, sin⁑(180∘+ΞΈ)=sin⁑θ\sin (180^{\circ} + \theta) = \sin \theta, tan⁑(180∘+ΞΈ)=tan⁑θ\tan (180^{\circ} + \theta) = \tan \theta, sin⁑(βˆ’ΞΈ)=βˆ’sin⁑θ\sin (-\theta) = -\sin \theta, cos⁑θ⋅sin⁑θ=12sin⁑2ΞΈ\cos \theta \cdot \sin \theta = \frac{1}{2} \sin 2\theta, and sin⁑(180∘+ΞΈ)=sin⁑θ\sin (180^{\circ} + \theta) = \sin \theta.

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

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

  • cos⁑(180∘+ΞΈ)=βˆ’cos⁑θ\cos (180^{\circ} + \theta) = -\cos \theta
  • sin⁑(180∘+ΞΈ)=sin⁑θ\sin (180^{\circ} + \theta) = \sin \theta
  • tan⁑(180∘+ΞΈ)=tan⁑θ\tan (180^{\circ} + \theta) = \tan \theta
  • sin⁑(βˆ’ΞΈ)=βˆ’sin⁑θ\sin (-\theta) = -\sin \theta
  • cos⁑θ⋅sin⁑θ=12sin⁑2ΞΈ\cos \theta \cdot \sin \theta = \frac{1}{2} \sin 2\theta
  • sin⁑(180∘+ΞΈ)=sin⁑θ\sin (180^{\circ} + \theta) = \sin \theta

Q: How can I apply these trigonometric identities in simplifying expressions?

A: To apply these trigonometric identities in simplifying expressions, you can follow these steps:

  1. Identify the trigonometric functions involved in the expression.
  2. Look for trigonometric identities that can be applied to simplify the expression.
  3. Use the trigonometric identities to rewrite the expression in a simpler form.
  4. Repeat the process until the expression is fully simplified.

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

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

  • Not using the correct trigonometric identities.
  • Not applying the trigonometric identities correctly.
  • Not simplifying the expression fully.
  • Not checking the final simplified expression for errors.

Conclusion

In this article, we answered some frequently asked questions related to the simplification of the expression cos⁑225βˆ˜β‹…sin⁑(βˆ’135∘)βˆ’sin⁑330∘tan⁑225∘\frac{\cos 225^{\circ} \cdot \sin \left(-135^{\circ}\right) - \sin 330^{\circ}}{\tan 225^{\circ}}. We provided some common trigonometric identities used in simplifying expressions and some common mistakes to avoid when simplifying expressions. We hope this article helps you in simplifying expressions and understanding trigonometric identities.