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 equation 4cos(x) + 5 = 6sec(x), we need to simplify it first. We know that sec(x) is the reciprocal of cos(x), so we can rewrite it as sec(x) = 1/cos(x).

Now let's substitute 1/cos(x) in place of sec(x) in the equation:

4cos(x) + 5 = 6(1/cos(x))

Next, let's multiply both sides of the equation by cos(x) to eliminate the fraction:

4cos(x)cos(x) + 5cos(x) = 6

Simplifying further:

4cos^2(x) + 5cos(x) - 6 = 0

This equation is a quadratic equation. To solve it, we can factor it or use the quadratic formula. Let's use factoring:

(4cos(x) - 1)(cos(x) + 6) = 0

Setting each factor equal to zero gives us two separate equations:

1) 4cos(x) - 1 = 0
2) cos(x) + 6 = 0

Solving equation 1:

4cos(x) - 1 = 0
4cos(x) = 1
cos(x) = 1/4

To find the angle whose cosine is 1/4, we can take the inverse cosine (also known as arccos or cos^(-1)) of 1/4. Using a calculator, we find that the angle is approximately 75.52 degrees.

Now let's solve equation 2:

cos(x) + 6 = 0
cos(x) = -6

However, cosine values only fall between -1 and 1, so there is no solution for equation 2. Therefore, we only have one solution, which is approximately x = 75.52 degrees.

To find the nearest minute, we can round to the nearest whole number since there are 60 minutes in a degree. Rounding 75.52 to the nearest whole number gives us 76 degrees.

Therefore, the solution to the equation 4cos(x) + 5 = 6sec(x), to the nearest minute, over the domain 0 degrees ≤ x ≤ 360 degrees, is x = 76 degrees.