Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- (1+x)/(x-7)>=0
- (x^2-6*x)/(x^2+6*x+9)<0
- 5^x+1+5^x<6
- (0,16x+0,3)/(exp((20x)/(20+x))*(-0,16x+0,7))>0
- Graphing y =:
- cos(6x)
- Integral of d{x}:
-
cos(6x)
-
-1/2
- Derivative of:
- cos(6x)
- -1/2
- Limit of the function:
- -1/2
-
Identical expressions
- cos(6x)<- one / two
- co sinus of e of (6x) less than minus 1 divide by 2
- co sinus of e of (6x) less than minus one divide by two
- cos6x<-1/2
- cos(6x)<-1 divide by 2
-
Similar expressions
- (x-(1/2))^x-(5/2)<0
- cos6*x>=-1/2
- cos(2x)>-(1/2)
- (1/2)^(x-2)-(1/2)^x<=3
- cos(6x)<+1/2
- cos(6x-1)>0.5
- cos6x≥-1/2
- (4-(1/2)^x/x-1)>0
cos(6x)<-1/2 inequation
The solution
Detail solution
Given the inequality:
$$\cos{\left(6 x \right)} < - \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(6 x \right)} = - \frac{1}{2}$$
Solve:
Given the equation
$$\cos{\left(6 x \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$6 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$6 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$6 x = \pi n + \frac{2 \pi}{3}$$
$$6 x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$6$$
$$x_{1} = \frac{\pi n}{6} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{18}$$
$$x_{1} = \frac{\pi n}{6} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{18}$$
This roots
$$x_{1} = \frac{\pi n}{6} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{18}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{6} + \frac{\pi}{9}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{6} - \frac{1}{10} + \frac{\pi}{9}$$
substitute to the expression
$$\cos{\left(6 x \right)} < - \frac{1}{2}$$
$$\cos{\left(6 \left(\frac{\pi n}{6} - \frac{1}{10} + \frac{\pi}{9}\right) \right)} < - \frac{1}{2}$$
but
Then
$$x < \frac{\pi n}{6} + \frac{\pi}{9}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{\pi n}{6} + \frac{\pi}{9} \wedge x < \frac{\pi n}{6} - \frac{\pi}{18}$$
$$\cos{\left(6 x \right)} < - \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(6 x \right)} = - \frac{1}{2}$$
Solve:
Given the equation
$$\cos{\left(6 x \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$6 x = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$6 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$6 x = \pi n + \frac{2 \pi}{3}$$
$$6 x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$6$$
$$x_{1} = \frac{\pi n}{6} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{18}$$
$$x_{1} = \frac{\pi n}{6} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{18}$$
This roots
$$x_{1} = \frac{\pi n}{6} + \frac{\pi}{9}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{18}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{6} + \frac{\pi}{9}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{6} - \frac{1}{10} + \frac{\pi}{9}$$
substitute to the expression
$$\cos{\left(6 x \right)} < - \frac{1}{2}$$
$$\cos{\left(6 \left(\frac{\pi n}{6} - \frac{1}{10} + \frac{\pi}{9}\right) \right)} < - \frac{1}{2}$$
/ 3 pi \
-sin|- - + -- + pi*n| < -1/2
\ 5 6 / but
/ 3 pi \
-sin|- - + -- + pi*n| > -1/2
\ 5 6 / Then
$$x < \frac{\pi n}{6} + \frac{\pi}{9}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{\pi n}{6} + \frac{\pi}{9} \wedge x < \frac{\pi n}{6} - \frac{\pi}{18}$$
_____
/ \
-------ο-------ο-------
x1 x2
Solving inequality on a graph
Rapid solution
[src]
/ / / /2*pi\\ \ / / /pi\\ \ \ | | |sin|----|| / _________________________\| | |sin|--|| / _____________________\| | | | | \ 9 /| | / 2/2*pi\ 2/2*pi\ || | | \9 /| | / 2/pi\ 2/pi\ || | And|x < -I*|I*atan|---------| + log| / cos |----| + sin |----| ||, -I*|I*atan|-------| + log| / cos |--| + sin |--| || < x| | | | /2*pi\| \\/ \ 9 / \ 9 / /| | | /pi\| \\/ \9 / \9 / /| | | | |cos|----|| | | |cos|--|| | | \ \ \ \ 9 // / \ \ \9 // / /
$$x < - i \left(\log{\left(\sqrt{\sin^{2}{\left(\frac{2 \pi}{9} \right)} + \cos^{2}{\left(\frac{2 \pi}{9} \right)}} \right)} + i \operatorname{atan}{\left(\frac{\sin{\left(\frac{2 \pi}{9} \right)}}{\cos{\left(\frac{2 \pi}{9} \right)}} \right)}\right) \wedge - i \left(\log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{9} \right)} + \cos^{2}{\left(\frac{\pi}{9} \right)}} \right)} + i \operatorname{atan}{\left(\frac{\sin{\left(\frac{\pi}{9} \right)}}{\cos{\left(\frac{\pi}{9} \right)}} \right)}\right) < x$$
(x < -i*(i*atan(sin(2*pi/9)/cos(2*pi/9)) + log(sqrt(cos(2*pi/9)^2 + sin(2*pi/9)^2))))∧(-i*(i*atan(sin(pi/9)/cos(pi/9)) + log(sqrt(cos(pi/9)^2 + sin(pi/9)^2))) < x)