Mister Exam
sin(x)+sqrt(3*cos(x))>1 inequation
The solution
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__________ sin(x) + \/ 3*cos(x) > 1
$$\sqrt{3 \cos{\left(x \right)}} + \sin{\left(x \right)} > 1$$
sqrt(3*cos(x)) + sin(x) > 1
Detail solution
Given the inequality:
$$\sqrt{3 \cos{\left(x \right)}} + \sin{\left(x \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3 \cos{\left(x \right)}} + \sin{\left(x \right)} = 1$$
Solve:
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
This roots
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
$$x_{1} = \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10}$$
=
$$2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sqrt{3 \cos{\left(x \right)}} + \sin{\left(x \right)} > 1$$
$$\sin{\left(2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10} \right)} + \sqrt{3 \cos{\left(2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10} \right)}} > 1$$
Then
$$x < 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} \wedge x < \frac{\pi}{2}$$
$$\sqrt{3 \cos{\left(x \right)}} + \sin{\left(x \right)} > 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{3 \cos{\left(x \right)}} + \sin{\left(x \right)} = 1$$
Solve:
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
This roots
$$x_{2} = 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
$$x_{1} = \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10}$$
=
$$2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10}$$
substitute to the expression
$$\sqrt{3 \cos{\left(x \right)}} + \sin{\left(x \right)} > 1$$
$$\sin{\left(2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10} \right)} + \sqrt{3 \cos{\left(2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} - \frac{1}{10} \right)}} > 1$$
____________________________________________________
/ / / / 2/3\\\ / / / 2/3\\\
/ | |3 ___ |3 ___ / ___\ ||| | |3 ___ |3 ___ / ___\ |||
___ / | 1 |\/ 2 *\\/ 2 - \1 + \/ 3 / /|| | 1 |\/ 2 *\\/ 2 - \1 + \/ 3 / /||
\/ 3 * / cos|- -- + 2*atan|------------------------------|| + sin|- -- + 2*atan|------------------------------|| > 1
/ | 10 | ___________ || | 10 | ___________ ||
/ | | 3 / ___ || | | 3 / ___ ||
\/ \ \ 2*\/ 1 + \/ 3 // \ \ 2*\/ 1 + \/ 3 //
Then
$$x < 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > 2 \operatorname{atan}{\left(\frac{\sqrt[3]{2} \left(- \left(1 + \sqrt{3}\right)^{\frac{2}{3}} + \sqrt[3]{2}\right)}{2 \sqrt[3]{1 + \sqrt{3}}} \right)} \wedge x < \frac{\pi}{2}$$
_____
/ \
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x2 x1
Solving inequality on a graph