Mister Exam
sin(x)+sqrt(3)*cos(x)<0 inequation
The solution
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___ sin(x) + \/ 3 *cos(x) < 0
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} < 0$$
sin(x) + sqrt(3)*cos(x) < 0
Detail solution
Given the inequality:
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = 0$$
transform:
$$\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} = - \sqrt{3}$$
or
$$\tan{\left(x \right)} = - \sqrt{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(\sqrt{3} \right)}$$
Or
$$x = \pi n + \frac{\pi}{3}$$
, where n - is a integer
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{1} = \pi n + \frac{\pi}{3}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{3}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{3}$$
substitute to the expression
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} < 0$$
$$\sin{\left(\pi n - \frac{1}{10} + \frac{\pi}{3} \right)} + \sqrt{3} \cos{\left(\pi n - \frac{1}{10} + \frac{\pi}{3} \right)} < 0$$
the solution of our inequality is:
$$x < \pi n + \frac{\pi}{3}$$
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} = 0$$
transform:
$$\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} = - \sqrt{3}$$
or
$$\tan{\left(x \right)} = - \sqrt{3}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(\sqrt{3} \right)}$$
Or
$$x = \pi n + \frac{\pi}{3}$$
, where n - is a integer
$$x_{1} = \pi n + \frac{\pi}{3}$$
$$x_{1} = \pi n + \frac{\pi}{3}$$
This roots
$$x_{1} = \pi n + \frac{\pi}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{3}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{3}$$
substitute to the expression
$$\sin{\left(x \right)} + \sqrt{3} \cos{\left(x \right)} < 0$$
$$\sin{\left(\pi n - \frac{1}{10} + \frac{\pi}{3} \right)} + \sqrt{3} \cos{\left(\pi n - \frac{1}{10} + \frac{\pi}{3} \right)} < 0$$
___ / 1 pi \ / 1 pi \
\/ 3 *cos|- -- + -- + pi*n| + sin|- -- + -- + pi*n| < 0
\ 10 3 / \ 10 3 / the solution of our inequality is:
$$x < \pi n + \frac{\pi}{3}$$
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Solving inequality on a graph
Rapid solution
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/2*pi 5*pi\ And|---- < x, x < ----| \ 3 3 /
$$\frac{2 \pi}{3} < x \wedge x < \frac{5 \pi}{3}$$
(2*pi/3 < x)∧(x < 5*pi/3)
Rapid solution 2
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2*pi 5*pi (----, ----) 3 3
$$x\ in\ \left(\frac{2 \pi}{3}, \frac{5 \pi}{3}\right)$$
x in Interval.open(2*pi/3, 5*pi/3)