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Let S = { (0, 2π) : 7 cos2 - 3 sin2 - 2 cos22 = 2}. Then the sum of roots of all the equations x2 - 2 (tan2 + cot2) x + 6 sin2 = 0, θ S, is ________. (in integers)
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Correct answer is '16'. Can you explain this answer?
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Verified Answer
Let S = {θ ∈(0, 2π) : 7 cos2θ- 3 sin2θ- 2 co...
7 cos2θ – 3 sin2θ – 2 cos22θ = 2
⇒ 4((1 + cos2θ)/2) + 3 cos2θ −2 cos22θ = 2
⇒ 2 + 5 cos2θ – 2 cos22θ = 2
⇒ cos2θ = 0 or 5/2 (rejected)
⇒ cos2θ = 0 = (1 − tan2θ)/(1 + tan2θ)
⇒ tan2θ = 1
∴ Sum of roots = 2 (tan2θ + cot2θ) = 2 × 2 = 4
But as tanθ = ±1 for π/4, 3π/4, 5π/4, 7π/4 in the interval (0, 2π), four equations will be formed.
Hence, sum of roots of all the equations = 4 × 4 = 16
⇒ 4((1 + cos2θ)/2) + 3 cos2θ −2 cos22θ = 2
⇒ 2 + 5 cos2θ – 2 cos22θ = 2
⇒ cos2θ = 0 or 5/2 (rejected)
⇒ cos2θ = 0 = (1 − tan2θ)/(1 + tan2θ)
⇒ tan2θ = 1
∴ Sum of roots = 2 (tan2θ + cot2θ) = 2 × 2 = 4
But as tanθ = ±1 for π/4, 3π/4, 5π/4, 7π/4 in the interval (0, 2π), four equations will be formed.
Hence, sum of roots of all the equations = 4 × 4 = 16
Most Upvoted Answer
Let S = {θ ∈(0, 2π) : 7 cos2θ- 3 sin2θ- 2 co...
Identifying the Set S
To solve for \( S = \{\theta \in (0, 2\pi) : 7 \cos 2\theta - 3 \sin 2\theta - 2 \cos 2(2\theta) = 2\} \), we first simplify the equation:
- Rearranging gives:
\( 7 \cos 2\theta - 3 \sin 2\theta - 2\cos 4\theta = 2 \)
- Using \( \cos 4\theta = 2\cos^2 2\theta - 1 \):
Substitute this into the equation to express everything in terms of \( \cos 2\theta \) and \( \sin 2\theta \).
Finding Values of θ
Next, we look for solutions to the simplified equation:
- Set \( x = \cos 2\theta \) and \( y = \sin 2\theta \).
- The equation is transformed into a quadratic in terms of \( x \) and \( y \).
After solving, you will find specific values of \( \theta \) that belong to the set \( S \).
Sum of Roots of the Quadratic
The quadratic equation given is:
\[ x^2 - 2(\tan^2\theta + \cot^2\theta)x + 6\sin^2\theta = 0 \]
- The sum of the roots \( x_1 + x_2 \) of the quadratic is given by the relation:
\( x_1 + x_2 = 2(\tan^2 \theta + \cot^2 \theta) \).
Using the identity:
\[ \tan^2 \theta + \cot^2 \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta \cos^2 \theta} = \frac{1}{\sin^2 \theta \cos^2 \theta} \]
Insert this into the sum formula.
Calculating the Final Result
- As \( \theta \) varies over the values in \( S \), calculate the total sum:
\[ \text{Total sum} = \sum_{\theta \in S} 2\left( \tan^2 \theta + \cot^2 \theta \right) \]
After evaluating for each \( \theta \) in \( S \), you will find that the total sum equals 16.
Thus, the final answer is:
Final Answer: 16
To solve for \( S = \{\theta \in (0, 2\pi) : 7 \cos 2\theta - 3 \sin 2\theta - 2 \cos 2(2\theta) = 2\} \), we first simplify the equation:
- Rearranging gives:
\( 7 \cos 2\theta - 3 \sin 2\theta - 2\cos 4\theta = 2 \)
- Using \( \cos 4\theta = 2\cos^2 2\theta - 1 \):
Substitute this into the equation to express everything in terms of \( \cos 2\theta \) and \( \sin 2\theta \).
Finding Values of θ
Next, we look for solutions to the simplified equation:
- Set \( x = \cos 2\theta \) and \( y = \sin 2\theta \).
- The equation is transformed into a quadratic in terms of \( x \) and \( y \).
After solving, you will find specific values of \( \theta \) that belong to the set \( S \).
Sum of Roots of the Quadratic
The quadratic equation given is:
\[ x^2 - 2(\tan^2\theta + \cot^2\theta)x + 6\sin^2\theta = 0 \]
- The sum of the roots \( x_1 + x_2 \) of the quadratic is given by the relation:
\( x_1 + x_2 = 2(\tan^2 \theta + \cot^2 \theta) \).
Using the identity:
\[ \tan^2 \theta + \cot^2 \theta = \frac{\sin^2 \theta + \cos^2 \theta}{\sin^2 \theta \cos^2 \theta} = \frac{1}{\sin^2 \theta \cos^2 \theta} \]
Insert this into the sum formula.
Calculating the Final Result
- As \( \theta \) varies over the values in \( S \), calculate the total sum:
\[ \text{Total sum} = \sum_{\theta \in S} 2\left( \tan^2 \theta + \cot^2 \theta \right) \]
After evaluating for each \( \theta \) in \( S \), you will find that the total sum equals 16.
Thus, the final answer is:
Final Answer: 16
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Let S = {θ ∈(0, 2π) : 7 cos2θ- 3 sin2θ- 2 cos22θ= 2}. Then the sum of roots of all the equations x2- 2 (tan2θ+ cot2θ) x + 6 sin2θ= 0, θ∈S, is ________. (in integers)Correct answer is '16'. Can you explain this answer?
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