find, to the nearest minute, the solution set of 4 cos x + 5 = 6 sec x over the domain 0 degrees which is less than or eaual to x which is less than 360 degrees
To solve the given equation 4 cos x + 5 = 6 sec x, we need to find the values of x that satisfy this equation in the domain 0 degrees ≤ x < 360 degrees.
Let's simplify the equation using trigonometric identities:
First, we know that sec x = 1/cos x, so we can substitute it into the equation:
4 cos x + 5 = 6 (1/cos x)
Next, to eliminate the fraction, we can multiply both sides of the equation by cos x:
(4 cos x + 5) cos x = 6
Expanding the left side:
4 cos^2 x + 5 cos x = 6
Rearranging the equation:
4 cos^2 x + 5 cos x - 6 = 0
Now, let's factorize this quadratic equation:
(4 cos x - 3)(cos x + 2) = 0
Setting each factor equal to zero and solving for x:
1) 4 cos x - 3 = 0
4 cos x = 3
cos x = 3/4
Using inverse cosine (cos^(-1)) to find the angle:
x = cos^(-1)(3/4)
2) cos x + 2 = 0
cos x = -2
However, the range of the cosine function is -1 ≤ cos x ≤ 1, so there are no solutions in this case.
Now, let's find the value of x using inverse cosine:
x = cos^(-1)(3/4)
Using a calculator or trigonometric table, we find:
x = 41.41 degrees or 318.59 degrees
Since we are looking for solutions within the domain 0 degrees ≤ x < 360 degrees, the solution set is:
x = 41.41 degrees and x = 318.59 degrees to the nearest minute.