PART 3: DEDUCTION AND APPLICATION:Do Not Use A Calculator When Answering Part 3.3.1 Apply The Compound Angle Expansion Sin ⁡ ( X + Y ) = Sin ⁡ X Cos ⁡ Y + Cos ⁡ X Sin ⁡ Y \sin (x+y) = \sin X \cos Y + \cos X \sin Y Sin ( X + Y ) = Sin X Cos Y + Cos X Sin Y To Sin ⁡ 2 X \sin 2x Sin 2 X And Simplify Your Answer:$[ \begin{aligned} \sin 2x

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3.3.1 Applying Compound Angle Expansion to Simplify sin⁡2x\sin 2x

In this section, we will apply the compound angle expansion formula to simplify the expression sin⁡2x\sin 2x. The compound angle expansion formula states that sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y. We will use this formula to rewrite sin⁡2x\sin 2x in a simplified form.

Understanding the Compound Angle Expansion Formula

The compound angle expansion formula is a fundamental concept in trigonometry that allows us to express the sine of a sum of two angles in terms of the sines and cosines of the individual angles. This formula is given by:

sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y

This formula can be used to simplify expressions involving the sine of a sum of two angles.

Applying the Compound Angle Expansion Formula to sin⁡2x\sin 2x

To simplify sin⁡2x\sin 2x, we can use the compound angle expansion formula by substituting x+yx+y with 2x2x. This gives us:

sin⁡2x=sin⁡(x+x)\sin 2x = \sin (x+x)

Using the compound angle expansion formula, we can rewrite this expression as:

sin⁡2x=sin⁡xcos⁡x+cos⁡xsin⁡x\sin 2x = \sin x \cos x + \cos x \sin x

Simplifying the Expression

We can simplify the expression further by combining like terms. The two terms sin⁡xcos⁡x\sin x \cos x and cos⁡xsin⁡x\cos x \sin x are identical, so we can combine them to get:

sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x

This is the simplified form of sin⁡2x\sin 2x.

Conclusion

In this section, we applied the compound angle expansion formula to simplify the expression sin⁡2x\sin 2x. We used the formula to rewrite sin⁡2x\sin 2x in a simplified form, which is 2sin⁡xcos⁡x2 \sin x \cos x. This result is a fundamental identity in trigonometry and is used extensively in various mathematical and scientific applications.

Discussion

The compound angle expansion formula is a powerful tool in trigonometry that allows us to simplify expressions involving the sine of a sum of two angles. By applying this formula, we can rewrite complex expressions in a simpler form, making it easier to work with them. In this section, we saw how the compound angle expansion formula can be used to simplify the expression sin⁡2x\sin 2x.

Key Takeaways

  • The compound angle expansion formula is given by sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y.
  • This formula can be used to simplify expressions involving the sine of a sum of two angles.
  • By applying the compound angle expansion formula, we can rewrite sin⁡2x\sin 2x in a simplified form, which is 2sin⁡xcos⁡x2 \sin x \cos x.

Further Reading

For more information on the compound angle expansion formula and its applications, see the following resources:

Practice Problems

Try the following practice problems to test your understanding of the compound angle expansion formula:

  • Simplify the expression sin⁡(x+y)\sin (x+y) using the compound angle expansion formula.
  • Use the compound angle expansion formula to simplify the expression cos⁡(x+y)\cos (x+y).
  • Show that the compound angle expansion formula is true for the expression sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y.
    PART 3: DEDUCTION AND APPLICATION =====================================

3.3.1 Applying Compound Angle Expansion to Simplify sin⁡2x\sin 2x

Q&A: Compound Angle Expansion and Simplifying sin⁡2x\sin 2x

In this section, we will answer some frequently asked questions about the compound angle expansion formula and its application to simplifying sin⁡2x\sin 2x.

Q: What is the compound angle expansion formula?

A: The compound angle expansion formula is a fundamental concept in trigonometry that allows us to express the sine of a sum of two angles in terms of the sines and cosines of the individual angles. The formula is given by:

sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y

Q: How do I apply the compound angle expansion formula to simplify sin⁡2x\sin 2x?

A: To simplify sin⁡2x\sin 2x, we can use the compound angle expansion formula by substituting x+yx+y with 2x2x. This gives us:

sin⁡2x=sin⁡(x+x)\sin 2x = \sin (x+x)

Using the compound angle expansion formula, we can rewrite this expression as:

sin⁡2x=sin⁡xcos⁡x+cos⁡xsin⁡x\sin 2x = \sin x \cos x + \cos x \sin x

Q: Why do we get 2sin⁡xcos⁡x2 \sin x \cos x as the simplified form of sin⁡2x\sin 2x?

A: We get 2sin⁡xcos⁡x2 \sin x \cos x as the simplified form of sin⁡2x\sin 2x because the two terms sin⁡xcos⁡x\sin x \cos x and cos⁡xsin⁡x\cos x \sin x are identical. When we combine like terms, we get:

sin⁡2x=2sin⁡xcos⁡x\sin 2x = 2 \sin x \cos x

Q: What are some common applications of the compound angle expansion formula?

A: The compound angle expansion formula has many applications in trigonometry and other mathematical and scientific fields. Some common applications include:

  • Simplifying expressions involving the sine of a sum of two angles
  • Finding the sine and cosine of a sum of two angles
  • Solving trigonometric equations involving the sine and cosine functions
  • Modeling periodic phenomena in physics and engineering

Q: How can I practice using the compound angle expansion formula?

A: You can practice using the compound angle expansion formula by trying the following exercises:

  • Simplify the expression sin⁡(x+y)\sin (x+y) using the compound angle expansion formula.
  • Use the compound angle expansion formula to simplify the expression cos⁡(x+y)\cos (x+y).
  • Show that the compound angle expansion formula is true for the expression sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y.

Q: What are some common mistakes to avoid when using the compound angle expansion formula?

A: Some common mistakes to avoid when using the compound angle expansion formula include:

  • Forgetting to substitute the correct values for xx and yy in the formula.
  • Not combining like terms correctly.
  • Not checking the validity of the formula for different values of xx and yy.

Conclusion

In this section, we answered some frequently asked questions about the compound angle expansion formula and its application to simplifying sin⁡2x\sin 2x. We hope that this Q&A article has been helpful in clarifying any doubts you may have had about the compound angle expansion formula.

Discussion

The compound angle expansion formula is a powerful tool in trigonometry that allows us to simplify expressions involving the sine of a sum of two angles. By applying this formula, we can rewrite complex expressions in a simpler form, making it easier to work with them. In this section, we saw how the compound angle expansion formula can be used to simplify the expression sin⁡2x\sin 2x.

Key Takeaways

  • The compound angle expansion formula is given by sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y.
  • This formula can be used to simplify expressions involving the sine of a sum of two angles.
  • By applying the compound angle expansion formula, we can rewrite sin⁡2x\sin 2x in a simplified form, which is 2sin⁡xcos⁡x2 \sin x \cos x.

Further Reading

For more information on the compound angle expansion formula and its applications, see the following resources:

Practice Problems

Try the following practice problems to test your understanding of the compound angle expansion formula:

  • Simplify the expression sin⁡(x+y)\sin (x+y) using the compound angle expansion formula.
  • Use the compound angle expansion formula to simplify the expression cos⁡(x+y)\cos (x+y).
  • Show that the compound angle expansion formula is true for the expression sin⁡(x+y)=sin⁡xcos⁡y+cos⁡xsin⁡y\sin (x+y) = \sin x \cos y + \cos x \sin y.