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if cosθ = √3/2
How many solutions does this equation have between -π and π ?
- a)2
- b)1
- c)3
- d)4
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
if cosθ = √3/2How many solutions does this equation have be...
The general solution of the given question is theta= 2nπ± π/6 but it is mentioned that they are lies between -π to π. So when we put n=0 we get theta =±π/6. And when we put n= 1 we get theta does not lies between -π to ÷π. So we get only two values of theta.
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if cosθ = √3/2How many solutions does this equation have be...
Understanding the Problem
To solve the equation where \(\cos \theta = \frac{\sqrt{3}}{2}\), we need to determine how many values of \(\theta\) exist within the interval \(-\pi\) to \(\pi\).
Cosine Function Properties
- The cosine function is periodic with a period of \(2\pi\).
- The values of \(\theta\) for which \(\cos \theta = \frac{\sqrt{3}}{2}\) are found at specific angles.
Angles Corresponding to Cosine Value
- The angles for which \(\cos \theta = \frac{\sqrt{3}}{2}\) are:
- \(\theta = \frac{\pi}{6}\) (in the first quadrant)
- \(\theta = -\frac{\pi}{6}\) (in the fourth quadrant)
Identifying Solutions in the Interval
- Now, we check the interval \(-\pi\) to \(\pi\):
- \(\theta = \frac{\pi}{6}\) is within the interval.
- \(\theta = -\frac{\pi}{6}\) is also within the interval.
Conclusion on Number of Solutions
- Therefore, there are **two valid solutions**: \(\frac{\pi}{6}\) and \(-\frac{\pi}{6}\).
- Hence, the correct answer is option **A**: 2 solutions.
This detailed analysis shows how the properties of the cosine function and the specific angles associated with its values lead to the conclusion that there are 2 solutions in the given interval.
To solve the equation where \(\cos \theta = \frac{\sqrt{3}}{2}\), we need to determine how many values of \(\theta\) exist within the interval \(-\pi\) to \(\pi\).
Cosine Function Properties
- The cosine function is periodic with a period of \(2\pi\).
- The values of \(\theta\) for which \(\cos \theta = \frac{\sqrt{3}}{2}\) are found at specific angles.
Angles Corresponding to Cosine Value
- The angles for which \(\cos \theta = \frac{\sqrt{3}}{2}\) are:
- \(\theta = \frac{\pi}{6}\) (in the first quadrant)
- \(\theta = -\frac{\pi}{6}\) (in the fourth quadrant)
Identifying Solutions in the Interval
- Now, we check the interval \(-\pi\) to \(\pi\):
- \(\theta = \frac{\pi}{6}\) is within the interval.
- \(\theta = -\frac{\pi}{6}\) is also within the interval.
Conclusion on Number of Solutions
- Therefore, there are **two valid solutions**: \(\frac{\pi}{6}\) and \(-\frac{\pi}{6}\).
- Hence, the correct answer is option **A**: 2 solutions.
This detailed analysis shows how the properties of the cosine function and the specific angles associated with its values lead to the conclusion that there are 2 solutions in the given interval.
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Community Answer
if cosθ = √3/2How many solutions does this equation have be...
The general solution of the given question is theta= 2nπ± π/6 but it is mentioned that they are lies between -π to π. So when we put n=0 we get theta =±π/6. And when we put n= 1 we get theta does not lies between -π to ÷π. So we get only two values of theta. So option A is correct.
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if cosθ = √3/2How many solutions does this equation have between -π and π ?a)2b)1c)3d)4Correct answer is option 'A'. Can you explain this answer?
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