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If 2sinθ/cos3θ = tan 270° - tan θ, find the value of θ?
- a)45°
- b)135°
- c)100°
- d)90°
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
If2sinθ/cos3θ = tan 270° - tan θ, find the value...
Formula:
sin2θ = 2sinθcosθ
sin(A - B) = sinAcosB - cosAsinB
sin2θ = 2sinθcosθ
sin(A - B) = sinAcosB - cosAsinB
Calculation:
Multiplying the left hand side with cosθ, we get-
Multiplying the left hand side with cosθ, we get-
sin2θ can also be written as sin(3θ - θ)
Applying sin(A-B) formula in the numerator part, we get-
⇒ tan3θ - tanθ
Equating LHS with RHS, we get-
⇒ tan3θ - tanθ = tan 270° - tan θ
⇒ 3θ = 270°
⇒ θ = 90°
∴ The value of θ is 90°
Applying sin(A-B) formula in the numerator part, we get-
⇒ tan3θ - tanθ
Equating LHS with RHS, we get-
⇒ tan3θ - tanθ = tan 270° - tan θ
⇒ 3θ = 270°
⇒ θ = 90°
∴ The value of θ is 90°
Most Upvoted Answer
If2sinθ/cos3θ = tan 270° - tan θ, find the value...
Understanding the Equation
We start with the equation:
2sinθ / cos3θ = tan 270° - tan θ
First, let's analyze the right side of the equation:
tan 270° Calculation
- tan 270° is undefined because it corresponds to a vertical asymptote in the tangent function.
- Therefore, we rewrite the equation as:
2sinθ / cos3θ = -tan θ
Rearranging the Equation
Now we can rearrange the equation:
- Multiply both sides by cos3θ to eliminate the fraction:
2sinθ = -tan θ * cos3θ
- Recall that tan θ = sinθ / cosθ, so we can substitute:
2sinθ = - (sinθ / cosθ) * cos3θ
Simplifying the Terms
- Multiplying both sides by cosθ gives us:
2sinθ * cosθ = -sinθ * cos3θ
- Dividing both sides by sinθ (assuming sinθ ≠ 0):
2cosθ = -cos3θ
Using the Cosine Triple Angle Identity
Now we can apply the cosine triple angle identity:
- cos3θ = 4cos^3θ - 3cosθ
Substituting this into the equation gives:
2cosθ = - (4cos^3θ - 3cosθ)
Final Rearrangement
- Rearranging yields:
4cos^3θ - 5cosθ = 0
- Factoring out cosθ:
cosθ(4cos^2θ - 5) = 0
This results in two cases:
1. cosθ = 0 → θ = 90° (which is the answer)
2. 4cos^2θ - 5 = 0 → cos^2θ = 5/4 (not valid since cos² cannot exceed 1)
Conclusion
The only valid solution is:
- θ = 90°
Thus, the answer is option 'D' (90°).
We start with the equation:
2sinθ / cos3θ = tan 270° - tan θ
First, let's analyze the right side of the equation:
tan 270° Calculation
- tan 270° is undefined because it corresponds to a vertical asymptote in the tangent function.
- Therefore, we rewrite the equation as:
2sinθ / cos3θ = -tan θ
Rearranging the Equation
Now we can rearrange the equation:
- Multiply both sides by cos3θ to eliminate the fraction:
2sinθ = -tan θ * cos3θ
- Recall that tan θ = sinθ / cosθ, so we can substitute:
2sinθ = - (sinθ / cosθ) * cos3θ
Simplifying the Terms
- Multiplying both sides by cosθ gives us:
2sinθ * cosθ = -sinθ * cos3θ
- Dividing both sides by sinθ (assuming sinθ ≠ 0):
2cosθ = -cos3θ
Using the Cosine Triple Angle Identity
Now we can apply the cosine triple angle identity:
- cos3θ = 4cos^3θ - 3cosθ
Substituting this into the equation gives:
2cosθ = - (4cos^3θ - 3cosθ)
Final Rearrangement
- Rearranging yields:
4cos^3θ - 5cosθ = 0
- Factoring out cosθ:
cosθ(4cos^2θ - 5) = 0
This results in two cases:
1. cosθ = 0 → θ = 90° (which is the answer)
2. 4cos^2θ - 5 = 0 → cos^2θ = 5/4 (not valid since cos² cannot exceed 1)
Conclusion
The only valid solution is:
- θ = 90°
Thus, the answer is option 'D' (90°).
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Community Answer
If2sinθ/cos3θ = tan 270° - tan θ, find the value...
Formula:
sin2θ = 2sinθcosθ
sin(A - B) = sinAcosB - cosAsinB
sin2θ = 2sinθcosθ
sin(A - B) = sinAcosB - cosAsinB
Calculation:
Multiplying the left hand side with cosθ, we get-
Multiplying the left hand side with cosθ, we get-
sin2θ can also be written as sin(3θ - θ)
Applying sin(A-B) formula in the numerator part, we get-
⇒ tan3θ - tanθ
Equating LHS with RHS, we get-
⇒ tan3θ - tanθ = tan 270° - tan θ
⇒ 3θ = 270°
⇒ θ = 90°
∴ The value of θ is 90°
Applying sin(A-B) formula in the numerator part, we get-
⇒ tan3θ - tanθ
Equating LHS with RHS, we get-
⇒ tan3θ - tanθ = tan 270° - tan θ
⇒ 3θ = 270°
⇒ θ = 90°
∴ The value of θ is 90°
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If2sinθ/cos3θ = tan 270° - tan θ, find the value of θ?a)45°b)135°c)100°d)90°Correct answer is option 'D'. Can you explain this answer?
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