Given Equation
The problem states that:
- sin theta + cos theta = (sqrt(3) + 1)/2
Objective
We need to show that:
- sin theta = sqrt(3) * cos theta
Squaring Both Sides
To manipulate the given equation, square both sides:
- (sin theta + cos theta)^2 = [(sqrt(3) + 1)/2]^2
This expands to:
- sin^2 theta + 2sin theta cos theta + cos^2 theta = (3 + 2sqrt(3) + 1)/4
Using the Pythagorean identity sin^2 theta + cos^2 theta = 1:
- 1 + 2sin theta cos theta = (4 + 2sqrt(3))/4
Rearranging the Equation
Now, simplify the right side:
- 2sin theta cos theta = (4 + 2sqrt(3))/4 - 1
This simplifies to:
- 2sin theta cos theta = (2 + 2sqrt(3))/4
Thus:
- 2sin theta cos theta = (1 + sqrt(3))/2
Using the Double Angle Identity
We know that:
- sin(2theta) = 2sin theta cos theta
Thus, we have:
- sin(2theta) = (1 + sqrt(3))/2
Solving for sin theta
Now, to show that:
- sin theta = sqrt(3) * cos theta, we can express sin theta in terms of cos theta:
From the earlier expression, we can derive:
- sin theta/cos theta = sqrt(3)
This implies:
- sin theta = sqrt(3) * cos theta
Conclusion
Thus, we have shown that if sin theta + cos theta = (sqrt(3) + 1)/2, then indeed:
- sin theta = sqrt(3) * cos theta.