Simplify The Following Expressions:1. $ \frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}} $2. $ 8.6 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin

by ADMIN 234 views

Introduction

Trigonometry is a branch of mathematics that deals with the relationships between the sides and angles of triangles. It is a fundamental subject that has numerous applications in various fields, including physics, engineering, and navigation. In this article, we will focus on simplifying two trigonometric expressions using various trigonometric identities and formulas.

Simplifying the First Expression

The first expression is given by:

3cos⁡150∘⋅sin⁡270∘tan⁡(−45∘)⋅cos⁡600∘\frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}}

To simplify this expression, we need to use various trigonometric identities and formulas. Let's start by evaluating the individual components of the expression.

  • Evaluating the numerator: The numerator of the expression is given by 3cos⁡150∘⋅sin⁡270∘3 \cos 150^{\circ} \cdot \sin 270^{\circ}. We can use the fact that cos⁡(180∘−x)=−cos⁡x\cos (180^{\circ} - x) = -\cos x and sin⁡(180∘−x)=sin⁡x\sin (180^{\circ} - x) = \sin x to rewrite the expression as 3cos⁡(30∘)⋅sin⁡(90∘)3 \cos (30^{\circ}) \cdot \sin (90^{\circ}).
  • Evaluating the denominator: The denominator of the expression is given by tan⁡(−45∘)⋅cos⁡600∘\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}. We can use the fact that tan⁡(−x)=−tan⁡x\tan (-x) = -\tan x to rewrite the expression as −tan⁡(45∘)⋅cos⁡(60∘)-\tan (45^{\circ}) \cdot \cos (60^{\circ}).

Now that we have evaluated the individual components of the expression, we can simplify the expression by combining the numerator and denominator.

3cos⁡150∘⋅sin⁡270∘tan⁡(−45∘)⋅cos⁡600∘=3cos⁡(30∘)⋅sin⁡(90∘)−tan⁡(45∘)⋅cos⁡(60∘)\frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}} = \frac{3 \cos (30^{\circ}) \cdot \sin (90^{\circ})}{-\tan (45^{\circ}) \cdot \cos (60^{\circ})}

Using the fact that cos⁡(30∘)=32\cos (30^{\circ}) = \frac{\sqrt{3}}{2}, sin⁡(90∘)=1\sin (90^{\circ}) = 1, tan⁡(45∘)=1\tan (45^{\circ}) = 1, and cos⁡(60∘)=12\cos (60^{\circ}) = \frac{1}{2}, we can simplify the expression further.

3cos⁡150∘⋅sin⁡270∘tan⁡(−45∘)⋅cos⁡600∘=3⋅32⋅1−1⋅12\frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}} = \frac{3 \cdot \frac{\sqrt{3}}{2} \cdot 1}{-1 \cdot \frac{1}{2}}

Simplifying the expression further, we get:

3cos⁡150∘⋅sin⁡270∘tan⁡(−45∘)⋅cos⁡600∘=−6\frac{3 \cos 150^{\circ} \cdot \sin 270^{\circ}}{\tan \left(-45^{\circ}\right) \cdot \cos 600^{\circ}} = -6

Simplifying the Second Expression

The second expression is given by:

8.6tan⁡(180∘−x)⋅sin⁡(90∘+x)sin⁡(180∘−x)⋅cos⁡(90∘+x)8.6 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin \left(180^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right)}

To simplify this expression, we need to use various trigonometric identities and formulas. Let's start by evaluating the individual components of the expression.

  • Evaluating the numerator: The numerator of the expression is given by tan⁡(180∘−x)⋅sin⁡(90∘+x)\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right). We can use the fact that tan⁡(180∘−x)=−tan⁡x\tan (180^{\circ} - x) = -\tan x and sin⁡(90∘+x)=cos⁡x\sin (90^{\circ} + x) = \cos x to rewrite the expression as −tan⁡x⋅cos⁡x-\tan x \cdot \cos x.
  • Evaluating the denominator: The denominator of the expression is given by sin⁡(180∘−x)⋅cos⁡(90∘+x)\sin \left(180^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right). We can use the fact that sin⁡(180∘−x)=sin⁡x\sin (180^{\circ} - x) = \sin x and cos⁡(90∘+x)=−sin⁡x\cos (90^{\circ} + x) = -\sin x to rewrite the expression as sin⁡x⋅−sin⁡x\sin x \cdot -\sin x.

Now that we have evaluated the individual components of the expression, we can simplify the expression by combining the numerator and denominator.

8.6tan⁡(180∘−x)⋅sin⁡(90∘+x)sin⁡(180∘−x)⋅cos⁡(90∘+x)=8.6−tan⁡x⋅cos⁡xsin⁡x⋅−sin⁡x8.6 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin \left(180^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right)} = 8.6 \frac{-\tan x \cdot \cos x}{\sin x \cdot -\sin x}

Using the fact that tan⁡x=sin⁡xcos⁡x\tan x = \frac{\sin x}{\cos x}, we can simplify the expression further.

8.6tan⁡(180∘−x)⋅sin⁡(90∘+x)sin⁡(180∘−x)⋅cos⁡(90∘+x)=8.6−sin⁡xcos⁡x⋅cos⁡xsin⁡x⋅−sin⁡x8.6 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin \left(180^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right)} = 8.6 \frac{-\frac{\sin x}{\cos x} \cdot \cos x}{\sin x \cdot -\sin x}

Simplifying the expression further, we get:

8.6tan⁡(180∘−x)⋅sin⁡(90∘+x)sin⁡(180∘−x)⋅cos⁡(90∘+x)=8.6−sin⁡xsin⁡x⋅−sin⁡x8.6 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin \left(180^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right)} = 8.6 \frac{-\sin x}{\sin x \cdot -\sin x}

Using the fact that sin⁡x⋅−sin⁡x=−sin⁡2x\sin x \cdot -\sin x = -\sin^2 x, we can simplify the expression further.

8.6tan⁡(180∘−x)⋅sin⁡(90∘+x)sin⁡(180∘−x)⋅cos⁡(90∘+x)=8.6−sin⁡x−sin⁡2x8.6 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin \left(180^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right)} = 8.6 \frac{-\sin x}{-\sin^2 x}

Simplifying the expression further, we get:

8.6tan⁡(180∘−x)⋅sin⁡(90∘+x)sin⁡(180∘−x)⋅cos⁡(90∘+x)=8.6sin⁡x8.6 \frac{\tan \left(180^{\circ}-x\right) \cdot \sin \left(90^{\circ}+x\right)}{\sin \left(180^{\circ}-x\right) \cdot \cos \left(90^{\circ}+x\right)} = \frac{8.6}{\sin x}

Conclusion

In this article, we have simplified two trigonometric expressions using various trigonometric identities and formulas. The first expression was simplified to −6-6, and the second expression was simplified to 8.6sin⁡x\frac{8.6}{\sin x}. These simplifications demonstrate the importance of using trigonometric identities and formulas to simplify complex expressions.

Final Answer

The final answer to the first expression is: -6

Q: What are some common trigonometric identities that can be used to simplify expressions?

A: Some common trigonometric identities that can be used to simplify expressions include:

  • Pythagorean identities: sin⁡2x+cos⁡2x=1\sin^2 x + \cos^2 x = 1, tan⁡2x+1=sec⁡2x\tan^2 x + 1 = \sec^2 x, and 1+cot⁡2x=csc⁡2x1 + \cot^2 x = \csc^2 x
  • Sum and difference identities: sin⁡(A+B)=sin⁡Acos⁡B+cos⁡Asin⁡B\sin (A + B) = \sin A \cos B + \cos A \sin B, sin⁡(A−B)=sin⁡Acos⁡B−cos⁡Asin⁡B\sin (A - B) = \sin A \cos B - \cos A \sin B, cos⁡(A+B)=cos⁡Acos⁡B−sin⁡Asin⁡B\cos (A + B) = \cos A \cos B - \sin A \sin B, and cos⁡(A−B)=cos⁡Acos⁡B+sin⁡Asin⁡B\cos (A - B) = \cos A \cos B + \sin A \sin B
  • Double-angle and half-angle identities: sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x, cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x - \sin^2 x, tan⁡2x=2tan⁡x1−tan⁡2x\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}, and sin⁡x2=±1−cos⁡x2\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}

Q: How can I simplify a trigonometric expression that involves multiple trigonometric functions?

A: To simplify a trigonometric expression that involves multiple trigonometric functions, you can use various trigonometric identities and formulas. Here are some steps you can follow:

  1. Identify the trigonometric functions: Identify the trigonometric functions involved in the expression, such as sine, cosine, and tangent.
  2. Use trigonometric identities: Use trigonometric identities to simplify the expression. For example, you can use the Pythagorean identities to simplify expressions involving sine and cosine.
  3. Combine like terms: Combine like terms in the expression to simplify it further.
  4. Use algebraic manipulations: Use algebraic manipulations, such as factoring and canceling, to simplify the expression.

Q: How can I simplify a trigonometric expression that involves a trigonometric function with a negative angle?

A: To simplify a trigonometric expression that involves a trigonometric function with a negative angle, you can use the following trigonometric identities:

  • Negative angle identities: sin⁡(−x)=−sin⁡x\sin (-x) = -\sin x, cos⁡(−x)=cos⁡x\cos (-x) = \cos x, and tan⁡(−x)=−tan⁡x\tan (-x) = -\tan x
  • Even and odd identities: sin⁡(−x)=−sin⁡x\sin (-x) = -\sin x and cos⁡(−x)=cos⁡x\cos (-x) = \cos x

Q: How can I simplify a trigonometric expression that involves a trigonometric function with a multiple angle?

A: To simplify a trigonometric expression that involves a trigonometric function with a multiple angle, you can use the following trigonometric identities:

  • Double-angle identities: sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x, cos⁡2x=cos⁡2x−sin⁡2x\cos 2x = \cos^2 x - \sin^2 x, and tan⁡2x=2tan⁡x1−tan⁡2x\tan 2x = \frac{2 \tan x}{1 - \tan^2 x}
  • Half-angle identities: sin⁡x2=±1−cos⁡x2\sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}} and cos⁡x2=±1+cos⁡x2\cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}

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

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

  • Not using trigonometric identities: Failing to use trigonometric identities can make it difficult to simplify expressions.
  • Not combining like terms: Failing to combine like terms can make it difficult to simplify expressions.
  • Not using algebraic manipulations: Failing to use algebraic manipulations, such as factoring and canceling, can make it difficult to simplify expressions.
  • Not checking for errors: Failing to check for errors can lead to incorrect simplifications.

Q: How can I practice simplifying trigonometric expressions?

A: To practice simplifying trigonometric expressions, you can try the following:

  • Work through examples: Work through examples of trigonometric expressions and try to simplify them using trigonometric identities and formulas.
  • Use online resources: Use online resources, such as calculators and worksheets, to practice simplifying trigonometric expressions.
  • Take practice tests: Take practice tests to assess your ability to simplify trigonometric expressions.
  • Seek help: Seek help from a teacher or tutor if you are having trouble simplifying trigonometric expressions.