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The least period of the function sin π[x]/12 + cos (πx/4) + tan π[x]/3 is λ, then the value of 201λ must be (where [.] denotes the greatest integer function)
Correct answer is '4824'. Can you explain this answer?
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Understanding the Functions
To find the least period of the given function, we analyze each component:
- Greatest Integer Function: The function sin(π[x]/12) is periodic with a period of 24 (since it is based on the periodicity of the sine function and the greatest integer function).
- Cosine Function: The function cos(πx/4) has a period of 8 (as the standard cosine function has a period of 2π and we divide x by 4).
- Tangent Function: The function tan(π[x]/3) has a period of 6 (tangent has a period of π, and here it is scaled by 1/3).
Finding the Least Common Multiple
To determine the least period of the combined function, we need to find the least common multiple (LCM) of the individual periods:
- Period of sin(π[x]/12) = 24
- Period of cos(πx/4) = 8
- Period of tan(π[x]/3) = 6
Calculating LCM
- Prime factorization of each period:
- 24 = 2^3 * 3
- 8 = 2^3
- 6 = 2 * 3
- The LCM takes the highest power of each prime:
- LCM = 2^3 (from 24 or 8) * 3^1 (from 24 or 6) = 24
Therefore, the least period λ = 24.
Final Calculation
To find 201λ, we calculate:
201 * 24 = 4824.
Thus, the answer is:
Final Answer: 4824
To find the least period of the given function, we analyze each component:
- Greatest Integer Function: The function sin(π[x]/12) is periodic with a period of 24 (since it is based on the periodicity of the sine function and the greatest integer function).
- Cosine Function: The function cos(πx/4) has a period of 8 (as the standard cosine function has a period of 2π and we divide x by 4).
- Tangent Function: The function tan(π[x]/3) has a period of 6 (tangent has a period of π, and here it is scaled by 1/3).
Finding the Least Common Multiple
To determine the least period of the combined function, we need to find the least common multiple (LCM) of the individual periods:
- Period of sin(π[x]/12) = 24
- Period of cos(πx/4) = 8
- Period of tan(π[x]/3) = 6
Calculating LCM
- Prime factorization of each period:
- 24 = 2^3 * 3
- 8 = 2^3
- 6 = 2 * 3
- The LCM takes the highest power of each prime:
- LCM = 2^3 (from 24 or 8) * 3^1 (from 24 or 6) = 24
Therefore, the least period λ = 24.
Final Calculation
To find 201λ, we calculate:
201 * 24 = 4824.
Thus, the answer is:
Final Answer: 4824
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The least period of the functionsinπ[x]/12 + cos (πx/4) + tanπ[x]/3is λ, then the value of201λmust be (where [.] denotes the greatest integer function)Correct answer is '4824'. Can you explain this answer?
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