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.