ind, to the nearest minute, the solution set of 4 cos x + 5 = 6 sec x over the domain 0 degrees
To find the solution set of the equation 4 cos(x) + 5 = 6 sec(x) over the domain [0 degrees, 360 degrees], we need to simplify the equation and solve for x.
First, let's simplify the equation using the trigonometric identity:
sec(x) = 1/cos(x)
Substituting this into the equation:
4 cos(x) + 5 = 6/cos(x)
Now, we can multiply through by cos(x) to get rid of the denominator:
4 cos^2(x) + 5 cos(x) - 6 = 0
Let's solve this quadratic equation. To do this, we can factor the equation:
(2 cos(x) + 3)(2 cos(x) - 2) = 0
Setting each factor equal to zero:
2 cos(x) + 3 = 0 or 2 cos(x) - 2 = 0
Solving the first equation:
2 cos(x) = -3
cos(x) = -3/2
However, the range of cosine function is [-1, 1], so there are no solutions for this equation.
Now, solving the second equation:
2 cos(x) = 2
cos(x) = 1
From the unit circle or trigonometric table, we know that cos(x) equals 1 at 0 degrees.
Therefore, the only solution within the given domain [0 degrees, 360 degrees] is x = 0 degrees.
Thus, the solution set to the nearest minute is {0}.