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Choose the correct time period of the function sin ωt + cos ωt
- a)π/8ω
- b)2π/ω
- c)π/4ω
- d)π/2ω
Correct answer is option 'B'. Can you explain this answer?
Most Upvoted Answer
Choose the correct time period of the function sin ωt + cos&omeg...
Understanding the Function
The function given is sin(ωt) + cos(ωt). To determine the period of this function, we need to analyze its components.
Components of the Function
- The sine function, sin(ωt), has a standard period of 2π.
- The cosine function, cos(ωt), also has a period of 2π.
Since both functions share the same frequency, the overall period of their sum will also be 2π.
Calculating the Period
1. Period of Sine Function:
- The period is given by T = 2π/ω.
2. Period of Cosine Function:
- Similarly, the period is also T = 2π/ω.
Since both functions oscillate with the same frequency, the resultant function sin(ωt) + cos(ωt) will also have a period of 2π/ω.
Correct Option
Thus, the correct answer for the time period of the function sin(ωt) + cos(ωt) is:
- Option B: 2π/ω
This means that every 2π/ω seconds, the function will complete one full cycle.
Conclusion
In summary, the periodic behavior of both sine and cosine functions, when combined, maintains the same period of 2π/ω. Therefore, recognizing these fundamental properties allows us to confidently determine the period of the combined function.
The function given is sin(ωt) + cos(ωt). To determine the period of this function, we need to analyze its components.
Components of the Function
- The sine function, sin(ωt), has a standard period of 2π.
- The cosine function, cos(ωt), also has a period of 2π.
Since both functions share the same frequency, the overall period of their sum will also be 2π.
Calculating the Period
1. Period of Sine Function:
- The period is given by T = 2π/ω.
2. Period of Cosine Function:
- Similarly, the period is also T = 2π/ω.
Since both functions oscillate with the same frequency, the resultant function sin(ωt) + cos(ωt) will also have a period of 2π/ω.
Correct Option
Thus, the correct answer for the time period of the function sin(ωt) + cos(ωt) is:
- Option B: 2π/ω
This means that every 2π/ω seconds, the function will complete one full cycle.
Conclusion
In summary, the periodic behavior of both sine and cosine functions, when combined, maintains the same period of 2π/ω. Therefore, recognizing these fundamental properties allows us to confidently determine the period of the combined function.
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Community Answer
Choose the correct time period of the function sin ωt + cos&omeg...
x = sin ωt - cos ωt
⇒ x = 2 cos ωt = 2 sin(ωt + π/2)
[sin C - sin D = 2 cos( (C + D)/2 ) sin( (C - D)/2 ) ]
⇒ x = 2 cos ωt = 2 sin(ωt + π/2)
[sin C - sin D = 2 cos( (C + D)/2 ) sin( (C - D)/2 ) ]
On comparing this with the equation of SHM: x = A sin(ωt + φ)
⇒ A = 2
ω (Angular frequency)
⇒ T = 2π/ω
⇒ A = 2
ω (Angular frequency)
⇒ T = 2π/ω
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Question Description
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