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If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0?
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If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0...
Proof that (3cosα - 4cos³α) = 0 when sinα = 1/2
Given, sinα = 1/2
We need to prove that (3cosα - 4cos³α) = 0
Using Trigonometric Identity
We know that sin²α + cos²α = 1 (Trigonometric Identity)
Substituting sinα = 1/2 in this equation, we get
(1/2)² + cos²α = 1
Simplifying it, we get
cos²α = 3/4
Taking square root on both sides, we get
cosα = ±√3/2
Checking for cosα = √3/2
If cosα = √3/2, then
3cosα - 4cos³α = 3(√3/2) - 4(√3/2)³
= 3(√3/2) - 4(3√3/8)
= (3√3/2) - (3√3/2)
= 0
Hence, for cosα = √3/2, (3cosα - 4cos³α) = 0
Checking for cosα = -√3/2
If cosα = -√3/2, then
3cosα - 4cos³α = 3(-√3/2) - 4(-√3/2)³
= -3(√3/2) - 4(-27√3/8)
= -3(√3/2) + 27√3/2
= 12√3/2
This is not equal to 0.
Hence, for cosα = -√3/2, (3cosα - 4cos³α) ≠ 0
Conclusion
Thus, we have proved that (3cosα - 4cos³α) = 0 when sinα = 1/2 only for cosα = √3/2.
Given, sinα = 1/2
We need to prove that (3cosα - 4cos³α) = 0
Using Trigonometric Identity
We know that sin²α + cos²α = 1 (Trigonometric Identity)
Substituting sinα = 1/2 in this equation, we get
(1/2)² + cos²α = 1
Simplifying it, we get
cos²α = 3/4
Taking square root on both sides, we get
cosα = ±√3/2
Checking for cosα = √3/2
If cosα = √3/2, then
3cosα - 4cos³α = 3(√3/2) - 4(√3/2)³
= 3(√3/2) - 4(3√3/8)
= (3√3/2) - (3√3/2)
= 0
Hence, for cosα = √3/2, (3cosα - 4cos³α) = 0
Checking for cosα = -√3/2
If cosα = -√3/2, then
3cosα - 4cos³α = 3(-√3/2) - 4(-√3/2)³
= -3(√3/2) - 4(-27√3/8)
= -3(√3/2) + 27√3/2
= 12√3/2
This is not equal to 0.
Hence, for cosα = -√3/2, (3cosα - 4cos³α) ≠ 0
Conclusion
Thus, we have proved that (3cosα - 4cos³α) = 0 when sinα = 1/2 only for cosα = √3/2.
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If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0?
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If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0?.
If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sin alpha = 1/2 than prove that ( 3 cos alpha - 4 cos^3 alpha ) = 0?.
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