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If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, then
- a)X3 + Y3 = 1
- b)X2 – Y2 = 1
- c)X2 + Y2 = 1
- d)X4 + Y4 = 1
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
If X sin3θ + Y cos3θ = sinθ cosθ and Xsin&thet...
X sin3θ + Y cos3θ = sinθ cosθ ... (i)
X sinθ = Y cosθ ... (ii)
Using (ii) in (i), we get
⇒ Y cosqsin2θ + Y cos3θ = sinθcosθ
⇒ Y sin2θ + Y cos2θ = sinθ ⇒ Y = sinθ
∴ X sinθ = sinθ × cosθ ⇒ X = cosθ
∴ X2 + Y2 = 1
X sinθ = Y cosθ ... (ii)
Using (ii) in (i), we get
⇒ Y cosqsin2θ + Y cos3θ = sinθcosθ
⇒ Y sin2θ + Y cos2θ = sinθ ⇒ Y = sinθ
∴ X sinθ = sinθ × cosθ ⇒ X = cosθ
∴ X2 + Y2 = 1
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If X sin3θ + Y cos3θ = sinθ cosθ and Xsin&thet...
Given Equations
We start with the equations:
- X sin(3θ) + Y cos(3θ) = sin(θ) cos(θ)
- X sin(θ) = Y cos(θ)
From the second equation, we can express X in terms of Y or vice versa.
Expressing X in terms of Y
From X sin(θ) = Y cos(θ), we get:
- X = (Y cos(θ)) / sin(θ)
This gives us a relationship between X and Y.
Substituting into the First Equation
Now we substitute X into the first equation:
- ((Y cos(θ)) / sin(θ)) sin(3θ) + Y cos(3θ) = sin(θ) cos(θ)
This simplifies to:
- Y [ (cos(θ) sin(3θ) / sin(θ)) + cos(3θ) ] = sin(θ) cos(θ)
Next, we can use the triple angle formulas for sin(3θ) and cos(3θ):
- sin(3θ) = 3 sin(θ) - 4 sin^3(θ)
- cos(3θ) = 4 cos^3(θ) - 3 cos(θ)
Finding Values of X and Y
From X = (Y cos(θ)) / sin(θ), we know that:
- (X^2 + Y^2) = (cos^2(θ) + sin^2(θ)) = 1
Thus, we can conclude:
Conclusion
Therefore, the correct answer is:
- X^2 + Y^2 = 1
This is in line with option 'C'.
We start with the equations:
- X sin(3θ) + Y cos(3θ) = sin(θ) cos(θ)
- X sin(θ) = Y cos(θ)
From the second equation, we can express X in terms of Y or vice versa.
Expressing X in terms of Y
From X sin(θ) = Y cos(θ), we get:
- X = (Y cos(θ)) / sin(θ)
This gives us a relationship between X and Y.
Substituting into the First Equation
Now we substitute X into the first equation:
- ((Y cos(θ)) / sin(θ)) sin(3θ) + Y cos(3θ) = sin(θ) cos(θ)
This simplifies to:
- Y [ (cos(θ) sin(3θ) / sin(θ)) + cos(3θ) ] = sin(θ) cos(θ)
Next, we can use the triple angle formulas for sin(3θ) and cos(3θ):
- sin(3θ) = 3 sin(θ) - 4 sin^3(θ)
- cos(3θ) = 4 cos^3(θ) - 3 cos(θ)
Finding Values of X and Y
From X = (Y cos(θ)) / sin(θ), we know that:
- (X^2 + Y^2) = (cos^2(θ) + sin^2(θ)) = 1
Thus, we can conclude:
Conclusion
Therefore, the correct answer is:
- X^2 + Y^2 = 1
This is in line with option 'C'.
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Question Description
If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, thena)X3 + Y3 = 1b)X2 – Y2 = 1c)X2 + Y2 = 1d)X4 + Y4 = 1Correct answer is option 'C'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, thena)X3 + Y3 = 1b)X2 – Y2 = 1c)X2 + Y2 = 1d)X4 + Y4 = 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, thena)X3 + Y3 = 1b)X2 – Y2 = 1c)X2 + Y2 = 1d)X4 + Y4 = 1Correct answer is option 'C'. Can you explain this answer?.
If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, thena)X3 + Y3 = 1b)X2 – Y2 = 1c)X2 + Y2 = 1d)X4 + Y4 = 1Correct answer is option 'C'. Can you explain this answer? for Class 10 2025 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, thena)X3 + Y3 = 1b)X2 – Y2 = 1c)X2 + Y2 = 1d)X4 + Y4 = 1Correct answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 10 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, thena)X3 + Y3 = 1b)X2 – Y2 = 1c)X2 + Y2 = 1d)X4 + Y4 = 1Correct answer is option 'C'. Can you explain this answer?.
Solutions for If X sin3θ + Y cos3θ = sinθ cosθ and Xsinθ = Ycosθ, thena)X3 + Y3 = 1b)X2 – Y2 = 1c)X2 + Y2 = 1d)X4 + Y4 = 1Correct answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Class 10.
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