Find all values of x in the interval [0, 2π] that satisfy the inequality. (Enter your answer using interval notation.)
14 cos(x) + 7 > 0
cosx>-1/2
http://www.btinternet.com/~tang/gcserevision/mathematics/images/cosine.gif
To solve the inequality 14cos(x) + 7 > 0, we can start by subtracting 7 from both sides:
14cos(x) > -7
Next, divide both sides by 14:
cos(x) > -7/14
Simplifying further:
cos(x) > -1/2
To find all values of x in the interval [0, 2π] that satisfy this inequality, we need to consider the values of x for which the cosine function is greater than -1/2.
The unit circle can help us determine the values of x that satisfy this condition. The cosine is positive in the first and fourth quadrants, so we need to find the angles in these quadrants where the cosine is greater than -1/2.
In the first quadrant, the cosine is positive, but it is always greater than -1/2. Therefore, all values of x in the first quadrant satisfy the inequality.
In the fourth quadrant, the cosine is also positive. The angle x in the fourth quadrant can be found using the reference angle, which is the angle between the positive x-axis and the terminal side of the angle. The reference angle is given by:
reference angle = arccos(-1/2) ≈ 2.0944
Since the fourth quadrant is a reflection of the first quadrant, the value of x in the fourth quadrant can be found by subtracting the reference angle from 2π:
x = 2π - reference angle ≈ 4.1888
Therefore, the values of x in the interval [0, 2π] that satisfy the inequality are:
[0, 2π]
To find the values of x that satisfy the inequality 14 cos(x) + 7 > 0, we can follow these steps:
1. Subtract 7 from both sides of the inequality:
14 cos(x) > -7
2. Divide both sides of the inequality by 14:
cos(x) > (-7/14)
3. Simplify the right side:
cos(x) > -1/2
Now, we need to find the values of x in the interval [0, 2π] where the cosine function is greater than -1/2. To do this, we can refer to the unit circle and identify the quadrants where cos(x) is positive.
The cosine function is positive in the first and fourth quadrants. In these quadrants, the values of cos(x) are greater than -1/2.
In the first quadrant, the values of x are between 0 and π/2.
In the fourth quadrant, the values of x are between 3π/2 and 2π.
Therefore, the values of x that satisfy the inequality are in the intervals [0, π/2) and (3π/2, 2π]. We can combine these intervals using the union symbol ∪, resulting in the interval [0, π/2) ∪ (3π/2, 2π]. This is the answer in interval notation.