8-12SIN^2X=4COS^2X
SOLVE FOR X
change cos^2 to 1-sin^2, regroup, and solve the quadratic equation .
To solve the equation 8 - 12sin^2(x) = 4cos^2(x), we can use basic trigonometric identities. Here's the step-by-step solution:
Step 1: Rewrite the equation using the identity sin^2(x) + cos^2(x) = 1.
8 - 12sin^2(x) = 4(1 - sin^2(x))
Step 2: Distribute 4 to get rid of the parentheses on the right side of the equation.
8 - 12sin^2(x) = 4 - 4sin^2(x)
Step 3: Combine like terms on both sides of the equation.
-12sin^2(x) + 12sin^2(x) = 4 - 8sin^2(x)
Step 4: Simplify the equation by adding like terms.
-8sin^2(x) = 4
Step 5: Divide both sides of the equation by -8 to isolate sin^2(x).
-8sin^2(x)/-8 = 4/-8
sin^2(x) = -1/2
Step 6: Take the square root of both sides of the equation to solve for sin(x).
√(sin^2(x)) = √(-1/2)
|sin(x)| = √(-1/2)
Step 7: Since the range of sine function is between -1 and 1, there is no real solution for sin(x) equal to the square root of a negative number. Therefore, there is no solution for x in this equation.
To solve the equation 8 - 12sin^2(x) = 4cos^2(x) for x, we can use the trigonometric identity sin^2(x) + cos^2(x) = 1 to simplify the equation.
Step 1: Use the identity sin^2(x) + cos^2(x) = 1 to rewrite the equation:
8 - 12sin^2(x) = 4(1 - sin^2(x))
Step 2: Distribute the 4 to get rid of the parentheses:
8 - 12sin^2(x) = 4 - 4sin^2(x)
Step 3: Combine like terms:
8 - 12sin^2(x) + 4sin^2(x) = 4
Step 4: Simplify the equation:
8 - 8sin^2(x) = 4
Step 5: Subtract 8 from both sides of the equation:
-8sin^2(x) = -4
Step 6: Divide both sides of the equation by -8:
sin^2(x) = 1/2
Step 7: Take the square root of both sides of the equation:
sin(x) = sqrt(1/2)
Step 8: Find the possible values of x using the domain of the sine function:
x = π/4 + 2πn, x = 3π/4 + 2πn, where n is an integer.
Therefore, the solutions for x are x = π/4 + 2πn and x = 3π/4 + 2πn, where n is an integer.