4cos(x) 5=6sec(x)
To solve the equation 4cos(x) + 5 = 6sec(x), we can use the trigonometric identity that relates cosine and secant functions. The identity is:
sec(x) = 1/cos(x)
We can substitute this identity into the equation to get:
4cos(x) + 5 = 6/cos(x)
Now, we can multiply both sides of the equation by cos(x) to eliminate the denominator:
4cos^2(x) + 5cos(x) = 6
Rearranging the equation:
4cos^2(x) + 5cos(x) - 6 = 0
Now, we have a quadratic equation in terms of cos(x). We can solve this equation by factoring, completing the square, or using the quadratic formula.
Factoring the quadratic equation, we need to find two numbers that multiply to -24 and add up to 5. The numbers are +8 and -3. We can rewrite the equation as:
(4cos(x) - 3)(cos(x) + 2) = 0
Now, we set each factor equal to zero:
4cos(x) - 3 = 0 or cos(x) + 2 = 0
Solving the first equation:
4cos(x) = 3
cos(x) = 3/4
To find the solutions for cos(x) = 3/4, we can use the inverse cosine function (also known as arccosine) to find the angles:
x = arccos(3/4) + 2πn or x = -arccos(3/4) + 2πn
where n is an integer.
Now, solving the second equation:
cos(x) + 2 = 0
cos(x) = -2
Since the cosine function has a range of -1 to 1, there are no real solutions for cos(x) = -2.
Therefore, the solutions for the equation 4cos(x) + 5 = 6sec(x) are:
x = arccos(3/4) + 2πn or x = -arccos(3/4) + 2πn
where n is an integer.