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What is the value of cos 6X?
- a)1 – 2sin23X
- b)2sin23X – 1
- c)cos23X – 1
- d)1 – 2cos23X
Correct answer is option 'A'. Can you explain this answer?
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Most Upvoted Answer
What is the value of cos 6X?a)1 – 2sin23Xb)2sin23X – 1c)co...
Explanation:
Trigonometric Identity:
Cosine double angle formula:
cos 2θ = 1 - 2sin²θ
Using Double Angle Formula:
Given cos 6X, we can rewrite this as cos (2 * 3X).
Using the double angle formula, we get:
cos (2 * 3X) = 1 - 2sin²(3X)
Expressing sin(3X) in terms of sin(23X):
To further simplify, we can use the triple angle formula:
sin 3X = 3sinX - 4sin³X
Substitute sinX = sin (1/2 * 6X) into the formula:
sin 3X = 3sin(2 * 3X) - 4sin³(2 * 3X)
= 3sin(6X) - 4sin³(6X)
= 3sin(6X) - 4(3sin(6X) - 4sin³(6X))
= 3sin(6X) - 12sin(6X) + 16sin³(6X)
= -9sin(6X) + 16sin³(6X)
Substitute sin(3X) back into the equation:
1 - 2(9sin²(6X) - 16sin⁴(6X))
= 1 - 18sin²(6X) + 32sin⁴(6X)
Final Simplification:
1 - 2(1 - cos²(6X))
= 1 - 2 + 2cos²(6X)
= 2cos²(6X) - 1
Therefore, the value of cos 6X is 2cos²(6X) - 1, which matches option A.
Trigonometric Identity:
Cosine double angle formula:
cos 2θ = 1 - 2sin²θ
Using Double Angle Formula:
Given cos 6X, we can rewrite this as cos (2 * 3X).
Using the double angle formula, we get:
cos (2 * 3X) = 1 - 2sin²(3X)
Expressing sin(3X) in terms of sin(23X):
To further simplify, we can use the triple angle formula:
sin 3X = 3sinX - 4sin³X
Substitute sinX = sin (1/2 * 6X) into the formula:
sin 3X = 3sin(2 * 3X) - 4sin³(2 * 3X)
= 3sin(6X) - 4sin³(6X)
= 3sin(6X) - 4(3sin(6X) - 4sin³(6X))
= 3sin(6X) - 12sin(6X) + 16sin³(6X)
= -9sin(6X) + 16sin³(6X)
Substitute sin(3X) back into the equation:
1 - 2(9sin²(6X) - 16sin⁴(6X))
= 1 - 18sin²(6X) + 32sin⁴(6X)
Final Simplification:
1 - 2(1 - cos²(6X))
= 1 - 2 + 2cos²(6X)
= 2cos²(6X) - 1
Therefore, the value of cos 6X is 2cos²(6X) - 1, which matches option A.
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Community Answer
What is the value of cos 6X?a)1 – 2sin23Xb)2sin23X – 1c)co...
Cos2x=1-2sin^2x
replace 2x with 6y x=3y
cos6y=1-2sin^23y
replace 2x with 6y x=3y
cos6y=1-2sin^23y
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What is the value of cos 6X?a)1 – 2sin23Xb)2sin23X – 1c)cos23X – 1d)1 – 2cos23XCorrect answer is option 'A'. Can you explain this answer?
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