Mister Exam
-cos(2*x+pi/6)+cos(-2*x+pi/3)<=1 inequation
The solution
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/ pi\ / pi\
- cos|2*x + --| + cos|-2*x + --| <= 1
\ 6 / \ 3 /
$$\cos{\left(- 2 x + \frac{\pi}{3} \right)} - \cos{\left(2 x + \frac{\pi}{6} \right)} \leq 1$$
cos(-2*x + pi/3) - cos(2*x + pi/6) <= 1
Detail solution
Given the inequality:
$$\cos{\left(- 2 x + \frac{\pi}{3} \right)} - \cos{\left(2 x + \frac{\pi}{6} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(- 2 x + \frac{\pi}{3} \right)} - \cos{\left(2 x + \frac{\pi}{6} \right)} = 1$$
Solve:
$$x_{1} = \frac{\pi}{6}$$
$$x_{1} = \frac{\pi}{6}$$
This roots
$$x_{1} = \frac{\pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$\cos{\left(- 2 x + \frac{\pi}{3} \right)} - \cos{\left(2 x + \frac{\pi}{6} \right)} \leq 1$$
$$- \cos{\left(\frac{\pi}{6} + 2 \left(- \frac{1}{10} + \frac{\pi}{6}\right) \right)} + \cos{\left(- 2 \left(- \frac{1}{10} + \frac{\pi}{6}\right) + \frac{\pi}{3} \right)} \leq 1$$
the solution of our inequality is:
$$x \leq \frac{\pi}{6}$$
$$\cos{\left(- 2 x + \frac{\pi}{3} \right)} - \cos{\left(2 x + \frac{\pi}{6} \right)} \leq 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(- 2 x + \frac{\pi}{3} \right)} - \cos{\left(2 x + \frac{\pi}{6} \right)} = 1$$
Solve:
$$x_{1} = \frac{\pi}{6}$$
$$x_{1} = \frac{\pi}{6}$$
This roots
$$x_{1} = \frac{\pi}{6}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
=
$$- \frac{1}{10} + \frac{\pi}{6}$$
substitute to the expression
$$\cos{\left(- 2 x + \frac{\pi}{3} \right)} - \cos{\left(2 x + \frac{\pi}{6} \right)} \leq 1$$
$$- \cos{\left(\frac{\pi}{6} + 2 \left(- \frac{1}{10} + \frac{\pi}{6}\right) \right)} + \cos{\left(- 2 \left(- \frac{1}{10} + \frac{\pi}{6}\right) + \frac{\pi}{3} \right)} \leq 1$$
-sin(1/5) + cos(1/5) <= 1
the solution of our inequality is:
$$x \leq \frac{\pi}{6}$$
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