tan^2(x)=-3/2sec(x)
sin^2 x/cos^2 x = 3/(2cosx)
2 cosx sin^2 x =3 cos^2 x
2cosx(1-cos^2 x) = 3cos^2 x
2cosx - 2cos^3 x - 3cos^2x = 0
-cosx(2cos2x + 3cosx - 2) = 0
cosx(cosx + 2)(2cosx - 1) = 0
cosx = 0 OR cosx = -2, not possible OR cosx= 1/2
x = 90° or 270° or 60° or 300°
in radians:
x = π/2, 3π/2, π/6 , 5π/6
Or, reading more carefully, and noticing the "-" sign,
sin^2 x/cos^2 x = -3/(2cosx)
2 cosx sin^2 x = -3 cos^2 x
2cosx(1-cos^2 x) = -3cos^2 x
2cosx - 2cos^3 x + 3cos^2x = 0
-cosx(2cos^2x - 3cosx - 2) = 0
cosx(cosx - 2)(2cosx + 1) = 0
cosx = 0 OR cosx = 2, not possible OR cosx= -1/2
x = 90° or 270° or 120° or 240°
in radians:
x = π/2, 3π/2, 2π/3 , 4π/3
To solve the equation tan^2(x) = -3/2 sec(x), we can use some trigonometric identities to simplify and manipulate the equation.
First, let's rewrite the equation in terms of sine and cosine. Recall that tan(x) = sin(x)/cos(x) and sec(x) = 1/cos(x). We can substitute these values in the equation:
(sin^2(x)/cos^2(x)) = -3/(2cos(x))
Next, let's multiply both sides of the equation by cos^2(x):
sin^2(x) = -3cos(x)/2
Now, let's use the Pythagorean identity to replace sin^2(x):
1 - cos^2(x) = -3cos(x)/2
Rearranging this equation, we get:
2 - 2cos^2(x) = -3cos(x)
Bringing all terms to one side:
2cos^2(x) - 3cos(x) + 2 = 0
Now, we have a quadratic equation in terms of cos(x). To solve this equation, we can use factoring, completing the square, or the quadratic formula.
Factoring:
Since the equation is quadratic, we can try to factor it. However, this equation does not easily factorize.
Completing the square:
To complete the square, we need to move the constant term to the other side of the equation:
2cos^2(x) - 3cos(x) = -2
Now, let's halve the coefficient of the middle term (-3/2), square it, and add it to both sides of the equation:
2cos^2(x) - 3cos(x) + (-3/4)^2 = -2 + (-3/4)^2
2cos^2(x) - 3cos(x) + 9/16 = -2 + 9/16
2cos^2(x) - 3cos(x) + 9/16 = -32/16 + 9/16
2cos^2(x) - 3cos(x) + 9/16 = -23/16
Now, we can write the left side of the equation as a perfect square:
(√2cos(x) - 3/4)^2 = -23/16
At this point, we can stop because the equation has a square term equal to a negative value, which is not possible. Hence, the equation tan^2(x) = -3/2sec(x) has no real solutions.