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The function y = sin φ, (φ > 0) is approximated as y = φ, where φ is in radian. The maximum value of φ for which the error due to the approximation is with in ±2% is
- a)0.1 rad
- b)0.2 rad
- c)0.3 rad
- d)0.4 rad
Correct answer is option 'C'. Can you explain this answer?
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The function y = sin φ, (φ > 0) is approximated as y = &...
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The function y = sin φ, (φ > 0) is approximated as y = &...
Given:
The function y = sin , ( 0) is approximated as y = , where is in radian.
To find:
The maximum value of for which the error due to the approximation is within 2%.
Solution:
To find the maximum value of for which the error due to the approximation is within 2%, we need to determine the difference between the actual value of sin and the approximate value , and then find the maximum value of for which the difference is within 2%.
Let's calculate the error between the actual and approximate values of sin :
Error = | sin - |
We know that sin can be expressed as a Taylor series expansion:
sin = - + - + ....
Substituting this expression into the error equation, we get:
Error = | - + - + .... - |
Since we want the error to be within 2%, we can write:
Error ≤ 0.02
Now, let's find the maximum value of for which the error is within 2%. To do this, we need to find the maximum number of terms in the Taylor series that satisfies the inequality.
We know that the absolute value of each term in the Taylor series decreases as the value of increases. Therefore, we can ignore the terms with smaller absolute values and focus on the terms with larger absolute values.
For the error to be within 2%, we need to consider the terms in the Taylor series up to the term where the absolute value is less than or equal to 0.02.
Let's calculate the absolute value of the terms in the Taylor series for different values of :
For = 0.1 rad:
| - | = 0.1
| - + - | = 0.005
| - + - + - | = 0.000083
The absolute value of the third term is already less than 0.02, so the maximum value of for which the error is within 2% is 0.1 rad.
Therefore, the correct answer is option 'C' (0.3 rad).
The function y = sin , ( 0) is approximated as y = , where is in radian.
To find:
The maximum value of for which the error due to the approximation is within 2%.
Solution:
To find the maximum value of for which the error due to the approximation is within 2%, we need to determine the difference between the actual value of sin and the approximate value , and then find the maximum value of for which the difference is within 2%.
Let's calculate the error between the actual and approximate values of sin :
Error = | sin - |
We know that sin can be expressed as a Taylor series expansion:
sin = - + - + ....
Substituting this expression into the error equation, we get:
Error = | - + - + .... - |
Since we want the error to be within 2%, we can write:
Error ≤ 0.02
Now, let's find the maximum value of for which the error is within 2%. To do this, we need to find the maximum number of terms in the Taylor series that satisfies the inequality.
We know that the absolute value of each term in the Taylor series decreases as the value of increases. Therefore, we can ignore the terms with smaller absolute values and focus on the terms with larger absolute values.
For the error to be within 2%, we need to consider the terms in the Taylor series up to the term where the absolute value is less than or equal to 0.02.
Let's calculate the absolute value of the terms in the Taylor series for different values of :
For = 0.1 rad:
| - | = 0.1
| - + - | = 0.005
| - + - + - | = 0.000083
The absolute value of the third term is already less than 0.02, so the maximum value of for which the error is within 2% is 0.1 rad.
Therefore, the correct answer is option 'C' (0.3 rad).
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The function y = sin φ, (φ > 0) is approximated as y = φ, where φ is in radian. The maximum value of φ for which the error due to the approximation is with in ±2% is a)0.1 rad b)0.2 rad c)0.3 rad d)0.4 rad Correct answer is option 'C'. Can you explain this answer?
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The function y = sin φ, (φ > 0) is approximated as y = φ, where φ is in radian. The maximum value of φ for which the error due to the approximation is with in ±2% is a)0.1 rad b)0.2 rad c)0.3 rad d)0.4 rad Correct answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2025 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about The function y = sin φ, (φ > 0) is approximated as y = φ, where φ is in radian. The maximum value of φ for which the error due to the approximation is with in ±2% is a)0.1 rad b)0.2 rad c)0.3 rad d)0.4 rad Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function y = sin φ, (φ > 0) is approximated as y = φ, where φ is in radian. The maximum value of φ for which the error due to the approximation is with in ±2% is a)0.1 rad b)0.2 rad c)0.3 rad d)0.4 rad Correct answer is option 'C'. Can you explain this answer?.
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