To solve this equation, we can use the double angle identity for sine, which states that:
sin(2θ) = 2sinθcosθ
Given the equation:
2sin^2(7π/2 + x) = 5sinx + 4
We can rewrite sin^2(7π/2 + x) as:
sin^2(7π/2 + x) = 1 - cos^2(7π/2 + x)
Now, let's plug this into the equation:
2(1 - cos^2(7π/2 + x)) = 5sinx + 42 - 2cos^2(7π/2 + x) = 5sinx + 4-2cos^2(7π/2 + x) = 5sinx + 2cos^2(7π/2 + x) = -(5sinx + 2)/2
Now, we can use the double angle identity for sine to convert cos^2(7π/2 + x) into a sine function:
sin^2(7π/2 + x) = 1 - cos^2(7π/2 + x)sin^2(7π/2 + x) = 1 - (-(5sinx + 2)/2)sin^2(7π/2 + x) = 1 + (5sinx + 2)/2
Now, we have sin^2(7π/2 + x) in terms of sinx. We can further simplify this expression and solve for sinx.
To solve this equation, we can use the double angle identity for sine, which states that:
sin(2θ) = 2sinθcosθ
Given the equation:
2sin^2(7π/2 + x) = 5sinx + 4
We can rewrite sin^2(7π/2 + x) as:
sin^2(7π/2 + x) = 1 - cos^2(7π/2 + x)
Now, let's plug this into the equation:
2(1 - cos^2(7π/2 + x)) = 5sinx + 4
2 - 2cos^2(7π/2 + x) = 5sinx + 4
-2cos^2(7π/2 + x) = 5sinx + 2
cos^2(7π/2 + x) = -(5sinx + 2)/2
Now, we can use the double angle identity for sine to convert cos^2(7π/2 + x) into a sine function:
sin^2(7π/2 + x) = 1 - cos^2(7π/2 + x)
sin^2(7π/2 + x) = 1 - (-(5sinx + 2)/2)
sin^2(7π/2 + x) = 1 + (5sinx + 2)/2
Now, we have sin^2(7π/2 + x) in terms of sinx. We can further simplify this expression and solve for sinx.