Solve step by step
4sin^2x = 3cos^2x
4sin^2x = 3(1-sin^2x)
4sin^2x = 3 - 3sin^2x
7sin^2x = 3
sin^2x = 3/7
sinx = ±√(3/7), so x is in any of the 4 quadrants
In first quadrant I get x = 40.9 degrees
in II, x = 180 - 40.9
in III, x = 180+40.9
in IV, x = 360-40.9
To solve the equation 4sin^2x = 3cos^2x, we can use trigonometric identities to simplify it and find the possible values of x.
Step 1: Start by rewriting the equation using the identity sin^2x = 1 - cos^2x:
4(1 - cos^2x) = 3cos^2x
Step 2: Distribute 4 on the left side of the equation:
4 - 4cos^2x = 3cos^2x
Step 3: Combine like terms:
4 = 7cos^2x
Step 4: Divide both sides of the equation by 7:
cos^2x = 4/7
Step 5: Take the square root of both sides of the equation to solve for cosx:
cosx = ± √(4/7)
Step 6: Simplify the square root:
cosx = ± (2√7) / 7
Step 7: To find the values of x, use the inverse cosine function (cos^-1) of both sides:
x = cos^-1((2√7) / 7) or x = cos^-1(-(2√7) / 7)
Step 8: Use a calculator to find the decimal or radian values of x.
Note: The symbol "√" represents the square root, and the symbol "±" indicates that there are two possible solutions, one for the positive value and one for the negative value. The inverse cosine function returns the angle (x) for which the cosine is equal to the given value.