To solve this trigonometric equation, we can first simplify it:
2cos(2x) + 4cos(3π/2 - x) + 1 = 0
Recall that cos(3π/2 - x) = sin(x), so the equation becomes:
2cos(2x) + 4sin(x) + 1 = 0
Now, using the double angle formula for cosine:
2(2cos^2(x) - 1) + 4sin(x) + 1 = 0
4cos^2(x) - 2 + 4sin(x) + 1 = 0
4cos^2(x) + 4sin(x) - 1 = 0
Divide the entire equation by 4:
cos^2(x) + sin(x) - 1/4 = 0
Substitute sin(x) with √(1 - cos^2(x)):
cos^2(x) + √(1 - cos^2(x)) - 1/4 = 0
Let y = cos(x):
y^2 + √(1 - y^2) - 1/4 = 0
This is a quadratic equation that can be solved for y. Then, after finding the values of y, we can substitute back y = cos(x) to find the solutions for x.
To solve this trigonometric equation, we can first simplify it:
2cos(2x) + 4cos(3π/2 - x) + 1 = 0
Recall that cos(3π/2 - x) = sin(x), so the equation becomes:
2cos(2x) + 4sin(x) + 1 = 0
Now, using the double angle formula for cosine:
2(2cos^2(x) - 1) + 4sin(x) + 1 = 0
4cos^2(x) - 2 + 4sin(x) + 1 = 0
4cos^2(x) + 4sin(x) - 1 = 0
Divide the entire equation by 4:
cos^2(x) + sin(x) - 1/4 = 0
Substitute sin(x) with √(1 - cos^2(x)):
cos^2(x) + √(1 - cos^2(x)) - 1/4 = 0
Let y = cos(x):
y^2 + √(1 - y^2) - 1/4 = 0
This is a quadratic equation that can be solved for y. Then, after finding the values of y, we can substitute back y = cos(x) to find the solutions for x.