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
Question content area bottom
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 is an identity, we'll start by substituting with 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:
For our case, we have:
Thus, we can write:
Step 2: Substitute into the equation
Substituting this into our equation:
This simplifies to:
Conclusion
Therefore, we have shown that:
which confirms that the equation is indeed an identity.
Summary
The identity 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:
- Can you explain the sine of a sum identity further?
- How does this identity apply in solving trigonometric equations?
- What are some other identities related to sine and cosine?
- How can this identity be used in calculus?
- 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