Mister Exam
(sinx/4+cosx/4)²≤1/2 inequation
The solution
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2 /sin(x) cos(x)\ |------ + ------| <= 1/2 \ 4 4 /
$$\left(\frac{\sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{4}\right)^{2} \leq \frac{1}{2}$$
(sin(x)/4 + cos(x)/4)^2 <= 1/2
Detail solution
Given the inequality:
$$\left(\frac{\sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{4}\right)^{2} \leq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{\sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{4}\right)^{2} = \frac{1}{2}$$
Solve:
$$x_{1} = 2 \operatorname{atan}{\left(- \frac{1}{7} + \frac{\sqrt{2} \sqrt{1 - 4 \sqrt{3} i}}{7} + \frac{4 \sqrt{3} i}{7} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{7} - \frac{\sqrt{2} \sqrt{1 + 4 \sqrt{3} i}}{7} + \frac{4 \sqrt{3} i}{7} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{1}{7} + \frac{\sqrt{2} \sqrt{1 + 4 \sqrt{3} i}}{7} + \frac{4 \sqrt{3} i}{7} \right)}$$
$$x_{4} = - 2 \operatorname{atan}{\left(\frac{1}{7} - \frac{4 \sqrt{3} i}{7} + \frac{\sqrt{2} \sqrt{1 - 4 \sqrt{3} i}}{7} \right)}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$\left(\frac{\sin{\left(0 \right)}}{4} + \frac{\cos{\left(0 \right)}}{4}\right)^{2} \leq \frac{1}{2}$$
so the inequality is always executed
$$\left(\frac{\sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{4}\right)^{2} \leq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\frac{\sin{\left(x \right)}}{4} + \frac{\cos{\left(x \right)}}{4}\right)^{2} = \frac{1}{2}$$
Solve:
$$x_{1} = 2 \operatorname{atan}{\left(- \frac{1}{7} + \frac{\sqrt{2} \sqrt{1 - 4 \sqrt{3} i}}{7} + \frac{4 \sqrt{3} i}{7} \right)}$$
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{1}{7} - \frac{\sqrt{2} \sqrt{1 + 4 \sqrt{3} i}}{7} + \frac{4 \sqrt{3} i}{7} \right)}$$
$$x_{3} = - 2 \operatorname{atan}{\left(\frac{1}{7} + \frac{\sqrt{2} \sqrt{1 + 4 \sqrt{3} i}}{7} + \frac{4 \sqrt{3} i}{7} \right)}$$
$$x_{4} = - 2 \operatorname{atan}{\left(\frac{1}{7} - \frac{4 \sqrt{3} i}{7} + \frac{\sqrt{2} \sqrt{1 - 4 \sqrt{3} i}}{7} \right)}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0
$$\left(\frac{\sin{\left(0 \right)}}{4} + \frac{\cos{\left(0 \right)}}{4}\right)^{2} \leq \frac{1}{2}$$
1/16 <= 1/2
so the inequality is always executed
Solving inequality on a graph
Rapid solution
This inequality holds true always