Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- log(7)(15-x)<1
- 9*x-3^(x+2)14>0
- 3z⩽2z+4
- (x-2)*(x+3):(x-4)>0
- Integral of d{x}:
-
cos(8x)
-
Identical expressions
- cos(8x)< zero
- co sinus of e of (8x) less than 0
- co sinus of e of (8x) less than zero
- cos8x<0
cos(8x)<0 inequation
The solution
Detail solution
Given the inequality:
$$\cos{\left(8 x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(8 x \right)} = 0$$
Solve:
Given the equation
$$\cos{\left(8 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(8 x \right)} = 0$$
This equation is transformed to
$$8 x = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$8 x = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$8 x = \pi n + \frac{\pi}{2}$$
$$8 x = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Divide both parts of the equation by
$$8$$
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{16}$$
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{16}$$
This roots
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{16}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{8} + \frac{\pi}{16}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{8} - \frac{1}{10} + \frac{\pi}{16}$$
substitute to the expression
$$\cos{\left(8 x \right)} < 0$$
$$\cos{\left(8 \left(\frac{\pi n}{8} - \frac{1}{10} + \frac{\pi}{16}\right) \right)} < 0$$
one of the solutions of our inequality is:
$$x < \frac{\pi n}{8} + \frac{\pi}{16}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x > \frac{\pi n}{8} - \frac{\pi}{16}$$
$$\cos{\left(8 x \right)} < 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(8 x \right)} = 0$$
Solve:
Given the equation
$$\cos{\left(8 x \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\cos{\left(8 x \right)} = 0$$
This equation is transformed to
$$8 x = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$8 x = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
Or
$$8 x = \pi n + \frac{\pi}{2}$$
$$8 x = \pi n - \frac{\pi}{2}$$
, where n - is a integer
Divide both parts of the equation by
$$8$$
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{16}$$
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{16}$$
This roots
$$x_{1} = \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x_{2} = \frac{\pi n}{8} - \frac{\pi}{16}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{8} + \frac{\pi}{16}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{8} - \frac{1}{10} + \frac{\pi}{16}$$
substitute to the expression
$$\cos{\left(8 x \right)} < 0$$
$$\cos{\left(8 \left(\frac{\pi n}{8} - \frac{1}{10} + \frac{\pi}{16}\right) \right)} < 0$$
-sin(-4/5 + pi*n) < 0
one of the solutions of our inequality is:
$$x < \frac{\pi n}{8} + \frac{\pi}{16}$$
_____ _____
\ /
-------ο-------ο-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x < \frac{\pi n}{8} + \frac{\pi}{16}$$
$$x > \frac{\pi n}{8} - \frac{\pi}{16}$$
Solving inequality on a graph
Rapid solution
[src]
/ / / ___________ ___________ \ \ \ | | | / ___ /pi\ / ___ /pi\| | / / /pi\\ \ | | | |\/ 2 + \/ 2 *sin|--| + \/ 2 - \/ 2 *cos|--|| / _____________________\| | |sin|--|| / _____________________\| | | | | \16/ \16/| | / 2/pi\ 2/pi\ || | | \16/| | / 2/pi\ 2/pi\ || | And|x < -I*|I*atan|-----------------------------------------------| + log| / cos |--| + sin |--| ||, -I*|I*atan|-------| + log| / cos |--| + sin |--| || < x| | | | ___________ ___________ | \\/ \16/ \16/ /| | | /pi\| \\/ \16/ \16/ /| | | | | / ___ /pi\ / ___ /pi\| | | |cos|--|| | | | | |\/ 2 + \/ 2 *cos|--| - \/ 2 - \/ 2 *sin|--|| | \ \ \16// / | \ \ \ \16/ \16// / /
$$x < - i \left(\log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)} + i \operatorname{atan}{\left(\frac{\sqrt{\sqrt{2} + 2} \sin{\left(\frac{\pi}{16} \right)} + \sqrt{2 - \sqrt{2}} \cos{\left(\frac{\pi}{16} \right)}}{- \sqrt{2 - \sqrt{2}} \sin{\left(\frac{\pi}{16} \right)} + \sqrt{\sqrt{2} + 2} \cos{\left(\frac{\pi}{16} \right)}} \right)}\right) \wedge - i \left(\log{\left(\sqrt{\sin^{2}{\left(\frac{\pi}{16} \right)} + \cos^{2}{\left(\frac{\pi}{16} \right)}} \right)} + i \operatorname{atan}{\left(\frac{\sin{\left(\frac{\pi}{16} \right)}}{\cos{\left(\frac{\pi}{16} \right)}} \right)}\right) < x$$
(-i*(i*atan(sin(pi/16)/cos(pi/16)) + log(sqrt(cos(pi/16)^2 + sin(pi/16)^2))) < x)∧(x < -i*(i*atan((sqrt(2 + sqrt(2))*sin(pi/16) + sqrt(2 - sqrt(2))*cos(pi/16))/(sqrt(2 + sqrt(2))*cos(pi/16) - sqrt(2 - sqrt(2))*sin(pi/16))) + log(sqrt(cos(pi/16)^2 + sin(pi/16)^2))))