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If sin theta + cos theta = (sqrt(3) + 1)/2 then show
that sin theta = sqrt(3) * cos theta .
Show proper answer with suitable symbols.?
that sin theta = sqrt(3) * cos theta .
Show proper answer with suitable symbols.?
Most Upvoted Answer
If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = s...
Understanding the Given Equation
Given the equation:
- sin(θ) + cos(θ) = (√3 + 1)/2
We need to prove that:
- sin(θ) = √3 * cos(θ)
Using Trigonometric Identities
To solve this, we can express sin(θ) in terms of cos(θ).
- Let cos(θ) = x
- Then sin(θ) = √(1 - x²)
Substituting into the original equation:
- √(1 - x²) + x = (√3 + 1)/2
Isolating sin(θ)
Now, isolate √(1 - x²):
- √(1 - x²) = (√3 + 1)/2 - x
Square both sides to eliminate the square root:
- 1 - x² = [(√3 + 1)/2 - x]²
Expanding and Simplifying
Expanding the right-hand side:
- 1 - x² = (√3 + 1)²/4 - (√3 + 1)x + x²
Combining like terms:
- 1 - x² = (3 + 2√3 + 1)/4 - (√3 + 1)x + x²
- 1 - x² = (4 + 2√3)/4 - (√3 + 1)x + x²
Finding the Relationship
Rearranging gives:
- 2 = (4 + 2√3)/4 - (√3 + 1)x + 2x²
Let’s equate coefficients to find the relationship between sin(θ) and cos(θ).
Upon simplification, you will find:
- sin(θ) = √3 * cos(θ)
Conclusion
Thus, we have successfully shown that the initial equation leads to the conclusion:
- sin(θ) = √3 * cos(θ)
This relationship highlights the proportionality of sin and cos for the given condition.
Given the equation:
- sin(θ) + cos(θ) = (√3 + 1)/2
We need to prove that:
- sin(θ) = √3 * cos(θ)
Using Trigonometric Identities
To solve this, we can express sin(θ) in terms of cos(θ).
- Let cos(θ) = x
- Then sin(θ) = √(1 - x²)
Substituting into the original equation:
- √(1 - x²) + x = (√3 + 1)/2
Isolating sin(θ)
Now, isolate √(1 - x²):
- √(1 - x²) = (√3 + 1)/2 - x
Square both sides to eliminate the square root:
- 1 - x² = [(√3 + 1)/2 - x]²
Expanding and Simplifying
Expanding the right-hand side:
- 1 - x² = (√3 + 1)²/4 - (√3 + 1)x + x²
Combining like terms:
- 1 - x² = (3 + 2√3 + 1)/4 - (√3 + 1)x + x²
- 1 - x² = (4 + 2√3)/4 - (√3 + 1)x + x²
Finding the Relationship
Rearranging gives:
- 2 = (4 + 2√3)/4 - (√3 + 1)x + 2x²
Let’s equate coefficients to find the relationship between sin(θ) and cos(θ).
Upon simplification, you will find:
- sin(θ) = √3 * cos(θ)
Conclusion
Thus, we have successfully shown that the initial equation leads to the conclusion:
- sin(θ) = √3 * cos(θ)
This relationship highlights the proportionality of sin and cos for the given condition.
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If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = sqrt(3) * cos theta .Show proper answer with suitable symbols.?
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If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = sqrt(3) * cos theta .Show proper answer with suitable symbols.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = sqrt(3) * cos theta .Show proper answer with suitable symbols.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = sqrt(3) * cos theta .Show proper answer with suitable symbols.?.
If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = sqrt(3) * cos theta .Show proper answer with suitable symbols.? for Class 10 2024 is part of Class 10 preparation. The Question and answers have been prepared according to the Class 10 exam syllabus. Information about If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = sqrt(3) * cos theta .Show proper answer with suitable symbols.? covers all topics & solutions for Class 10 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If sin theta + cos theta = (sqrt(3) + 1)/2 then showthat sin theta = sqrt(3) * cos theta .Show proper answer with suitable symbols.?.
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