Simplify The Following:1. Sin ⁡ ( Θ − 45 ∘ ) + Sin ⁡ ( Θ + 45 ∘ \sin (\theta-45^{\circ})+\sin (\theta+45^{\circ} Sin ( Θ − 4 5 ∘ ) + Sin ( Θ + 4 5 ∘ ]2. Cos ⁡ ( Α + 60 ∘ ) − Cos ⁡ ( Α − 60 ∘ \cos (\alpha+60^{\circ})-\cos (\alpha-60^{\circ} Cos ( Α + 6 0 ∘ ) − Cos ( Α − 6 0 ∘ ]3. Sin ⁡ ( Β + 60 ∘ ) − Cos ⁡ ( Β − 30 ∘ \sin (\beta+60^{\circ})-\cos (\beta-30^{\circ} Sin ( Β + 6 0 ∘ ) − Cos ( Β − 3 0 ∘ ]4. $\cos (A+60^{\circ})-\sin

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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 trigonometric expressions, which is an essential skill for anyone studying trigonometry.

Simplifying Trigonometric Expressions

1. Simplifying sin⁡(θ−45∘)+sin⁡(θ+45∘)\sin (\theta-45^{\circ})+\sin (\theta+45^{\circ})

To simplify the expression sin⁡(θ−45∘)+sin⁡(θ+45∘)\sin (\theta-45^{\circ})+\sin (\theta+45^{\circ}), we can use the sum-to-product identity, which states that sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right).

Using this identity, we can rewrite the expression as:

sin⁡(θ−45∘)+sin⁡(θ+45∘)=2sin⁡((θ−45∘)+(θ+45∘)2)cos⁡((θ−45∘)−(θ+45∘)2)\sin (\theta-45^{\circ})+\sin (\theta+45^{\circ}) = 2\sin \left(\frac{(\theta-45^{\circ})+(\theta+45^{\circ})}{2}\right)\cos \left(\frac{(\theta-45^{\circ})-(\theta+45^{\circ})}{2}\right)

Simplifying the expression further, we get:

sin⁡(θ−45∘)+sin⁡(θ+45∘)=2sin⁡(2θ2)cos⁡(−90∘2)\sin (\theta-45^{\circ})+\sin (\theta+45^{\circ}) = 2\sin \left(\frac{2\theta}{2}\right)\cos \left(\frac{-90^{\circ}}{2}\right)

=2sin⁡θcos⁡(−45∘)= 2\sin \theta \cos (-45^{\circ})

Since cos⁡(−45∘)=cos⁡45∘\cos (-45^{\circ}) = \cos 45^{\circ}, we can rewrite the expression as:

sin⁡(θ−45∘)+sin⁡(θ+45∘)=2sin⁡θcos⁡45∘\sin (\theta-45^{\circ})+\sin (\theta+45^{\circ}) = 2\sin \theta \cos 45^{\circ}

=2sin⁡θ= \sqrt{2}\sin \theta

Therefore, the simplified expression is 2sin⁡θ\sqrt{2}\sin \theta.

2. Simplifying cos⁡(α+60∘)−cos⁡(α−60∘)\cos (\alpha+60^{\circ})-\cos (\alpha-60^{\circ})

To simplify the expression cos⁡(α+60∘)−cos⁡(α−60∘)\cos (\alpha+60^{\circ})-\cos (\alpha-60^{\circ}), we can use the difference-to-product identity, which states that cos⁡A−cos⁡B=−2sin⁡(A+B2)sin⁡(A−B2)\cos A - \cos B = -2\sin \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right).

Using this identity, we can rewrite the expression as:

cos⁡(α+60∘)−cos⁡(α−60∘)=−2sin⁡((α+60∘)+(α−60∘)2)sin⁡((α+60∘)−(α−60∘)2)\cos (\alpha+60^{\circ})-\cos (\alpha-60^{\circ}) = -2\sin \left(\frac{(\alpha+60^{\circ})+(\alpha-60^{\circ})}{2}\right)\sin \left(\frac{(\alpha+60^{\circ})-(\alpha-60^{\circ})}{2}\right)

Simplifying the expression further, we get:

cos⁡(α+60∘)−cos⁡(α−60∘)=−2sin⁡(2α2)sin⁡(120∘2)\cos (\alpha+60^{\circ})-\cos (\alpha-60^{\circ}) = -2\sin \left(\frac{2\alpha}{2}\right)\sin \left(\frac{120^{\circ}}{2}\right)

=−2sin⁡αsin⁡60∘= -2\sin \alpha \sin 60^{\circ}

Since sin⁡60∘=32\sin 60^{\circ} = \frac{\sqrt{3}}{2}, we can rewrite the expression as:

cos⁡(α+60∘)−cos⁡(α−60∘)=−3sin⁡α\cos (\alpha+60^{\circ})-\cos (\alpha-60^{\circ}) = -\sqrt{3}\sin \alpha

Therefore, the simplified expression is −3sin⁡α-\sqrt{3}\sin \alpha.

3. Simplifying sin⁡(β+60∘)−cos⁡(β−30∘)\sin (\beta+60^{\circ})-\cos (\beta-30^{\circ})

To simplify the expression sin⁡(β+60∘)−cos⁡(β−30∘)\sin (\beta+60^{\circ})-\cos (\beta-30^{\circ}), we can use the sum-to-product identity for sine and the difference-to-product identity for cosine.

Using the sum-to-product identity for sine, we can rewrite the expression as:

sin⁡(β+60∘)−cos⁡(β−30∘)=sin⁡(β+60∘)−cos⁡(β−30∘)\sin (\beta+60^{\circ})-\cos (\beta-30^{\circ}) = \sin (\beta+60^{\circ}) - \cos (\beta-30^{\circ})

=2sin⁡((β+60∘)+(β−30∘)2)cos⁡((β+60∘)−(β−30∘)2)−cos⁡(β−30∘)= 2\sin \left(\frac{(\beta+60^{\circ})+(\beta-30^{\circ})}{2}\right)\cos \left(\frac{(\beta+60^{\circ})-(\beta-30^{\circ})}{2}\right) - \cos (\beta-30^{\circ})

Simplifying the expression further, we get:

sin⁡(β+60∘)−cos⁡(β−30∘)=2sin⁡(2β2)cos⁡(90∘2)−cos⁡(β−30∘)\sin (\beta+60^{\circ})-\cos (\beta-30^{\circ}) = 2\sin \left(\frac{2\beta}{2}\right)\cos \left(\frac{90^{\circ}}{2}\right) - \cos (\beta-30^{\circ})

=2sin⁡βcos⁡45∘−cos⁡(β−30∘)= 2\sin \beta \cos 45^{\circ} - \cos (\beta-30^{\circ})

Since cos⁡45∘=22\cos 45^{\circ} = \frac{\sqrt{2}}{2}, we can rewrite the expression as:

sin⁡(β+60∘)−cos⁡(β−30∘)=2sin⁡β−cos⁡(β−30∘)\sin (\beta+60^{\circ})-\cos (\beta-30^{\circ}) = \sqrt{2}\sin \beta - \cos (\beta-30^{\circ})

Using the difference-to-product identity for cosine, we can rewrite the expression as:

sin⁡(β+60∘)−cos⁡(β−30∘)=2sin⁡β−cos⁡(β−30∘)\sin (\beta+60^{\circ})-\cos (\beta-30^{\circ}) = \sqrt{2}\sin \beta - \cos (\beta-30^{\circ})

=−2sin⁡((β−30∘)+(β+60∘)2)sin⁡((β−30∘)−(β+60∘)2)= -2\sin \left(\frac{(\beta-30^{\circ})+(\beta+60^{\circ})}{2}\right)\sin \left(\frac{(\beta-30^{\circ})-(\beta+60^{\circ})}{2}\right)

Simplifying the expression further, we get:

sin⁡(β+60∘)−cos⁡(β−30∘)=−2sin⁡(2β−30∘2)sin⁡(−90∘2)\sin (\beta+60^{\circ})-\cos (\beta-30^{\circ}) = -2\sin \left(\frac{2\beta-30^{\circ}}{2}\right)\sin \left(\frac{-90^{\circ}}{2}\right)

=−2sin⁡(β−15∘2)cos⁡45∘= -2\sin \left(\frac{\beta-15^{\circ}}{2}\right)\cos 45^{\circ}

=−2sin⁡(β−15∘2)= -\sqrt{2}\sin \left(\frac{\beta-15^{\circ}}{2}\right)

Therefore, the simplified expression is −2sin⁡(β−15∘2)-\sqrt{2}\sin \left(\frac{\beta-15^{\circ}}{2}\right).

4. Simplifying cos⁡(A+60∘)−sin⁡(A−30∘)\cos (A+60^{\circ})-\sin (A-30^{\circ})

To simplify the expression cos⁡(A+60∘)−sin⁡(A−30∘)\cos (A+60^{\circ})-\sin (A-30^{\circ}), we can use the difference-to-product identity for cosine and the sum-to-product identity for sine.

Using the difference-to-product identity for cosine, we can rewrite the expression as:

cos⁡(A+60∘)−sin⁡(A−30∘)=−2sin⁡((A+60∘)+(A−30∘)2)sin⁡((A+60∘)−(A−30∘)2)\cos (A+60^{\circ})-\sin (A-30^{\circ}) = -2\sin \left(\frac{(A+60^{\circ})+(A-30^{\circ})}{2}\right)\sin \left(\frac{(A+60^{\circ})-(A-30^{\circ})}{2}\right)

Simplifying the expression further, we get:

cos⁡(A+60∘)−sin⁡(A−30∘)=−2sin⁡(2A2)sin⁡(90∘2)\cos (A+60^{\circ})-\sin (A-30^{\circ}) = -2\sin \left(\frac{2A}{2}\right)\sin \left(\frac{90^{\circ}}{2}\right)

=−2sin⁡Asin⁡45∘= -2\sin A \sin 45^{\circ}

Since sin⁡45∘=22\sin 45^{\circ} = \frac{\sqrt{2}}{2}, we can rewrite the expression as:

cos⁡(A+60∘)−sin⁡(A−30∘)=−2sin⁡A\cos (A+60^{\circ})-\sin (A-30^{\circ}) = -\sqrt{2}\sin A

Using the sum-to-product identity for sine, we can rewrite the expression as:

cos⁡(A+60∘)−sin⁡(A−30∘)=−2sin⁡A\cos (A+60^{\circ})-\sin (A-30^{\circ}) = -\sqrt{2}\sin A

=2sin⁡(A+60∘+A−30∘2)cos⁡(A+60∘−A+30∘2)= 2\sin \left(\frac{A+60^{\circ}+A-30^{\circ}}{2}\right)\cos \left(\frac{A+60^{\circ}-A+30^{\circ}}{2}\right)

Simplifying the expression further, we get:

cos⁡(A+60∘)−sin⁡(A−30∘)=2sin⁡(2A+30∘2)cos⁡(90∘2)\cos (A+60^{\circ})-\sin (A-30^{\circ}) = 2\sin \left(\frac{2A+30^{\circ}}{2}\right)\cos \left(\frac{90^{\circ}}{2}\right)

=2sin⁡(A+15∘)cos⁡45∘= 2\sin \left(A+15^{\circ}\right)\cos 45^{\circ}

=2sin⁡(A+15∘)= \sqrt{2}\sin \left(A+15^{\circ}\right)

Therefore, the simplified expression is 2sin⁡(A+15∘)\sqrt{2}\sin \left(A+15^{\circ}\right).

Conclusion

Q&A: Simplifying Trigonometric Expressions

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 the sum-to-product identity, the difference-to-product identity, and the product-to-sum identity.

Q: How do I use the sum-to-product identity to simplify an expression? A: To use the sum-to-product identity, you need to identify the two terms in the expression that you want to simplify. Then, you can use the formula:

sin⁡A+sin⁡B=2sin⁡(A+B2)cos⁡(A−B2)\sin A + \sin B = 2\sin \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)

or

cos⁡A+cos⁡B=2cos⁡(A+B2)cos⁡(A−B2)\cos A + \cos B = 2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)

to simplify the expression.

Q: How do I use the difference-to-product identity to simplify an expression? A: To use the difference-to-product identity, you need to identify the two terms in the expression that you want to simplify. Then, you can use the formula:

sin⁡A−sin⁡B=2cos⁡(A+B2)sin⁡(A−B2)\sin A - \sin B = 2\cos \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)

or

cos⁡A−cos⁡B=−2sin⁡(A+B2)sin⁡(A−B2)\cos A - \cos B = -2\sin \left(\frac{A+B}{2}\right)\sin \left(\frac{A-B}{2}\right)

to simplify the expression.

Q: What is the product-to-sum identity, and how do I use it to simplify an expression? A: The product-to-sum identity is a formula that allows you to express the product of two trigonometric functions as a sum of two trigonometric functions. The formula is:

sin⁡Acos⁡B=12[sin⁡(A+B)+sin⁡(A−B)]\sin A \cos B = \frac{1}{2}[\sin (A+B) + \sin (A-B)]

or

cos⁡Acos⁡B=12[cos⁡(A+B)+cos⁡(A−B)]\cos A \cos B = \frac{1}{2}[\cos (A+B) + \cos (A-B)]

You can use this formula to simplify expressions that involve the product of two trigonometric functions.

Q: How do I choose which identity to use to simplify an expression? A: To choose which identity to use, you need to look at the expression and identify the terms that you want to simplify. Then, you can use the identity that is most applicable to the terms in the expression.

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 the correct identity
  • Not simplifying the expression fully
  • Not checking the expression for errors
  • Not using the correct trigonometric function (e.g. using sin⁡\sin instead of cos⁡\cos)

Q: How do I check my work when simplifying trigonometric expressions? A: To check your work, you can use the following steps:

  • Simplify the expression using the identity
  • Check the expression to make sure it is simplified fully
  • Check the expression to make sure it is correct
  • Use a calculator or other tool to check the expression

Conclusion

In this article, we have provided a comprehensive guide to simplifying trigonometric expressions using various identities. We have also provided a Q&A section to answer common questions and provide additional tips and resources. By following the steps outlined in this article, you can simplify complex trigonometric expressions and improve your understanding of trigonometry.

Additional Resources

  • Trigonometric identities: A list of common trigonometric identities and their formulas.
  • Trigonometric functions: A list of common trigonometric functions and their properties.
  • Trigonometry tutorials: A list of online tutorials and resources for learning trigonometry.
  • Trigonometry problems: A list of practice problems and exercises for practicing trigonometry.

Final Tips

  • Practice, practice, practice: The more you practice simplifying trigonometric expressions, the more comfortable you will become with the identities and formulas.
  • Use a calculator or other tool: A calculator or other tool can be a big help when simplifying trigonometric expressions.
  • Check your work: Always check your work to make sure it is correct and simplified fully.
  • Seek help: If you are having trouble simplifying a trigonometric expression, don't be afraid to seek help from a teacher, tutor, or online resource.