Determine all the possible values of x where 0 deg is more than or equal to x and x is more or equal to 360 deg such that
i'm not sure the solution, please correct it and how to find the x value,
2 tan x - 1 = cot x
solution:
(sec x -1)(sec x + 2) = 0
sec -1 = 0 , sec x + 2 = 0
sec x = 1 , sec x = -2
1/cos x = 1 , 1/cos x = -2
cos x = 1 or cos x = -2
then i don't know how to continue...
To solve the equation 2tan(x) - 1 = cot(x), you can use the identities tan(x) = sin(x)/cos(x) and cot(x) = cos(x)/sin(x). Substitute these values into the equation:
2(sin(x)/cos(x)) - 1 = cos(x)/sin(x)
Multiply through by cos(x) and sin(x):
2sin(x)sin(x) - cos(x)cos(x) = cos(x)
Using the Pythagorean identity sin^2(x) + cos^2(x) = 1, we can rewrite the equation as:
2sin(x)sin(x) - (1 - sin^2(x)) = cos(x)
Expand and simplify:
2sin^2(x) - 1 + sin^2(x) = cos(x)
Combine like terms:
3sin^2(x) - 1 = cos(x)
Rearranging the equation:
cos(x) = 3sin^2(x) - 1
Now, we can use the identity cos^2(x) + sin^2(x) = 1 to rewrite the equation:
1 - sin^2(x) = 3sin^2(x) - 1
Bring all the terms to one side:
4sin^2(x) - sin^2(x) - 1 = 0
Simplify:
3sin^2(x) - 1 = 0
To find the solutions for sin(x), we can solve this quadratic equation:
3sin^2(x) - 1 = 0
Multiply through by 1/3:
sin^2(x) = 1/3
Take the square root of both sides:
sin(x) = ± sqrt(1/3)
sin(x) = ± 1/sqrt(3)
Now, to find the possible values of x, we need to find the angles whose sine is equal to ± 1/sqrt(3). You can use a calculator or look up the values to find that these angles are 30 degrees and 150 degrees.
Therefore, the possible values of x where 0 degrees ≤ x ≤ 360 degrees are x = 30 degrees and x = 150 degrees.