Math Problem Statement

Verify that the equation is an identity.​ (Hint:

sine 2 xsin2xequals=sine left parenthesis x plus x right parenthesissin(x+x)​)

sine 2 xsin2xequals=2 sine x cosine x2sin xcos x

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Part 1

Substitute

2xequals=xplus+x

and apply the sine of a sum identity.

sine 2 xsin2x

equals=

sine left parenthesis x plus x right parenthesissin(x+x)

equals=

enter your response here

​(Do not​ simplify.)

Solution

To verify that the equation sin(2x)=sin(x+x)\sin(2x) = \sin(x + x) is an identity, we'll start by substituting 2x2x with x+xx + x and apply the sine of a sum identity.

Step 1: Use the sine of a sum identity

The sine of a sum identity states that:

sin(a+b)=sin(a)cos(b)+cos(a)sin(b)\sin(a + b) = \sin(a)\cos(b) + \cos(a)\sin(b)

For our case, we have:

a=xandb=xa = x \quad \text{and} \quad b = x

Thus, we can write:

sin(x+x)=sin(x)cos(x)+cos(x)sin(x)\sin(x + x) = \sin(x)\cos(x) + \cos(x)\sin(x)

Step 2: Substitute into the equation

Substituting this into our equation:

sin(2x)=sin(x+x)=sin(x)cos(x)+cos(x)sin(x)\sin(2x) = \sin(x + x) = \sin(x)\cos(x) + \cos(x)\sin(x)

This simplifies to:

sin(2x)=2sin(x)cos(x)\sin(2x) = 2\sin(x)\cos(x)

Conclusion

Therefore, we have shown that:

sin(2x)=2sin(x)cos(x)\sin(2x) = 2\sin(x)\cos(x)

which confirms that the equation is indeed an identity.

Summary

The identity sin(2x)=2sin(x)cos(x)\sin(2x) = 2\sin(x)\cos(x) holds true based on the substitution and the application of the sine of a sum identity.

Would you like more details or do you have any questions? Here are some follow-up questions:

  1. Can you explain the sine of a sum identity further?
  2. How does this identity apply in solving trigonometric equations?
  3. What are some other identities related to sine and cosine?
  4. How can this identity be used in calculus?
  5. Can you derive the cosine double angle identity in a similar way?

Tip: Remember to practice using trigonometric identities in different problems to strengthen your understanding!

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sine Function
Identities

Formulas

sin(2x) = 2sin(x)cos(x)
sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Theorems

Sine of a sum identity

Suitable Grade Level

Grades 9-12