Find all solutions of cos (x) + 1/2 sec (x) = -3/2 in the interval (2pi, 4pi)
(Leave your answers in exact form and enter them as a comma-separated list.)
cosx + (1/2)(1/cosx) = -3/2
times 2
2cosx + 1/cosx = -3
times cosx
2cos^2 x + 1 = -3cosx
2cos^2 x + 3cosx + 1 = 0
(2cosx + 1)(cosx + 1) = 0
cosx = -1/2 or cosx = -1
x = 120° or x = 240° or x = 180°
in radians:
x = 2π/3 , 4π/3 , π
BUT, you want the solutions in (2π,4π) so let's just look one period over
x = 2π/3 + 2π = 8π/3
x = 4π/3 + 2π = 10π/3
x = π + 2π = 3π
To solve the equation cos(x) + 1/2 sec(x) = -3/2 in the interval (2π, 4π), we can start by rewriting sec(x) as 1/cos(x).
The equation now becomes:
cos(x) + 1/2(1/cos(x)) = -3/2
To get rid of the denominator, we can multiply the entire equation by 2cos(x):
2cos(x)cos(x) + 2(1) = -3(2cos(x))
2cos^2(x) + 2 = -6cos(x)
Next, let's move all the terms to one side of the equation:
2cos^2(x) + 6cos(x) + 2 = 0
To solve this quadratic equation, we can use the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 2, b = 6, and c = 2. Substituting these values into the quadratic formula, we get:
x = (-6 ± √(6^2 - 4(2)(2))) / (2(2))
Simplifying inside the square root:
x = (-6 ± √(36 - 16)) / 4
x = (-6 ± √20) / 4
x = (-6 ± 2√5) / 4
Simplifying further:
x = (-3 ± √5) / 2
Since we are looking for solutions in the interval (2π, 4π), we need to find x values that satisfy:
2π < x < 4π
Substituting the solutions we found back into the original equation to check if they satisfy the equation:
For x = (-3 + √5) / 2:
cos((-3 + √5) / 2) + 1/2sec((-3 + √5) / 2) = -3/2
The same with x = (-3 - √5) / 2.
Unfortunately, analyzing these equations is tedious and complicated. Additionally, finding exact solutions in this interval is not straightforward. So for practical purposes, it would be better to use numerical methods, such as graphing or using a calculator, to find approximate solutions.
To find all solutions of the equation cos(x) + 1/2sec(x) = -3/2 in the interval (2π, 4π), we can follow these steps:
Step 1: Begin by substituting sec(x) with its reciprocal, which is 1/cos(x). The equation becomes cos(x) + 1/(2cos(x)) = -3/2.
Step 2: Simplify the equation by finding the common denominator. Multiply the second term, 1/(2cos(x)), by cos(x)/cos(x) to get cos(x)/(2cos(x)).
The equation now becomes:
cos(x) + cos(x)/(2cos(x)) = -3/2.
Step 3: Combine the two terms on the left side by finding a common denominator. The equation becomes (2cos(x) + cos(x))/(2cos(x)) = -3/2.
Step 4: Simplify the left side of the equation:
(3cos(x))/(2cos(x)) = -3/2.
The cos(x) term cancels out, leaving us with:
3/2 = -3/2.
Step 5: Since 3/2 is not equal to -3/2, there are no solutions for this equation in the given interval (2π, 4π).
Therefore, there are no solutions to the equation cos(x) + 1/2sec(x) = -3/2 in the interval (2π, 4π).