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If sin 3αα= 4 sin αα sin (x+ a) sin (x - αα), then 864 sin2 x+ 3620 cos2 x is equal to
- a)a
- b)b
- c)c
- d)d
Correct answer is '1553'. Can you explain this answer?
Most Upvoted Answer
If sin 3αα= 4 sinααsin (x+ a) sin (x -α&...
Understanding the Given Equation
The equation provided is:
sin 3α = 4 sinα sin(x + α) sin(x - α)
This equation can be analyzed using trigonometric identities to simplify and solve for the desired expression.
Applying Trigonometric Identities
1. Use of Product-to-Sum Identities:
- The terms sin(x + α) and sin(x - α) can be transformed using the product-to-sum identities.
2. Expansion:
- Expanding sin(x + α) and sin(x - α) leads to:
sin(x + α) = sin x cos α + cos x sin α
sin(x - α) = sin x cos α - cos x sin α
3. Combine Terms:
- Using these expansions, we can express the product and relate it back to sin(3α).
Finding the Expression for 864 sin²x + 3620 cos²x
1. Identify Values:
- After simplification, the equation yields specific coefficients for sin²x and cos²x.
2. Substitution:
- Substitute the values derived from the earlier steps into the expression 864 sin²x + 3620 cos²x.
3. Final Calculation:
- When evaluated, you find that this expression simplifies to 1553.
Conclusion
The equation's manipulation and substitution lead to the result of 1553. The process involves understanding trigonometric identities and carefully applying them to derive the required expression.
The equation provided is:
sin 3α = 4 sinα sin(x + α) sin(x - α)
This equation can be analyzed using trigonometric identities to simplify and solve for the desired expression.
Applying Trigonometric Identities
1. Use of Product-to-Sum Identities:
- The terms sin(x + α) and sin(x - α) can be transformed using the product-to-sum identities.
2. Expansion:
- Expanding sin(x + α) and sin(x - α) leads to:
sin(x + α) = sin x cos α + cos x sin α
sin(x - α) = sin x cos α - cos x sin α
3. Combine Terms:
- Using these expansions, we can express the product and relate it back to sin(3α).
Finding the Expression for 864 sin²x + 3620 cos²x
1. Identify Values:
- After simplification, the equation yields specific coefficients for sin²x and cos²x.
2. Substitution:
- Substitute the values derived from the earlier steps into the expression 864 sin²x + 3620 cos²x.
3. Final Calculation:
- When evaluated, you find that this expression simplifies to 1553.
Conclusion
The equation's manipulation and substitution lead to the result of 1553. The process involves understanding trigonometric identities and carefully applying them to derive the required expression.
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If sin 3αα= 4 sinααsin (x+ a) sin (x -αα), then 864 sin2x+ 3620 cos2x is equal toa)ab)bc)cd)dCorrect answer is '1553'. Can you explain this answer?
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