Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- 2*(x-1)-3*(x-2)<x
- (x+10)*1/2>x-2
- (x-1/4)(x-8)>=0
- (1/2)^(2x+1)<4
- Graphing y =:
- cos(6x)
- Integral of d{x}:
-
cos(6x)
- Derivative of:
- cos(6x)
-
Identical expressions
- cos(6x)<-(sqrt3/ two)
- co sinus of e of (6x) less than minus ( square root of 3 divide by 2)
- co sinus of e of (6x) less than minus ( square root of 3 divide by two)
- cos(6x)<-(√3/2)
- cos6x<-sqrt3/2
- cos(6x)<-(sqrt3 divide by 2)
-
Similar expressions
- cos6x>-1/2
- -12cos6xsin6x<3
- sinx/4<-sqrt(3)/2
- sin(x/4)<-sqrt3/2
- cos(6x)<+(sqrt3/2)
- sin^6(x)*cos^6(x)>=1/4
- cos6x<=√2/2
- cos(pi/4-3x)≤-sqrt3/2
- cos(x/2)<(-sqrt(3)/2)
cos(6x)<-(sqrt3/2) inequation
The solution
You have entered
[src]
___
-\/ 3
cos(6*x) < -------
2
$$\cos{\left(6 x \right)} < - \frac{\sqrt{3}}{2}$$
cos(6*x) < -sqrt(3)/2
Detail solution
Given the inequality:
$$\cos{\left(6 x \right)} < - \frac{\sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(6 x \right)} = - \frac{\sqrt{3}}{2}$$
Solve:
Given the equation
$$\cos{\left(6 x \right)} = - \frac{\sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$6 x = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$6 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
Or
$$6 x = \pi n + \frac{5 \pi}{6}$$
$$6 x = \pi n - \frac{\pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$6$$
$$x_{1} = \frac{\pi n}{6} + \frac{5 \pi}{36}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{36}$$
$$x_{1} = \frac{\pi n}{6} + \frac{5 \pi}{36}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{36}$$
This roots
$$x_{1} = \frac{\pi n}{6} + \frac{5 \pi}{36}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{36}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{6} + \frac{5 \pi}{36}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{6} - \frac{1}{10} + \frac{5 \pi}{36}$$
substitute to the expression
$$\cos{\left(6 x \right)} < - \frac{\sqrt{3}}{2}$$
$$\cos{\left(6 \left(\frac{\pi n}{6} - \frac{1}{10} + \frac{5 \pi}{36}\right) \right)} < - \frac{\sqrt{3}}{2}$$
but
Then
$$x < \frac{\pi n}{6} + \frac{5 \pi}{36}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{\pi n}{6} + \frac{5 \pi}{36} \wedge x < \frac{\pi n}{6} - \frac{\pi}{36}$$
$$\cos{\left(6 x \right)} < - \frac{\sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(6 x \right)} = - \frac{\sqrt{3}}{2}$$
Solve:
Given the equation
$$\cos{\left(6 x \right)} = - \frac{\sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$6 x = \pi n + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$6 x = \pi n - \pi + \operatorname{acos}{\left(- \frac{\sqrt{3}}{2} \right)}$$
Or
$$6 x = \pi n + \frac{5 \pi}{6}$$
$$6 x = \pi n - \frac{\pi}{6}$$
, where n - is a integer
Divide both parts of the equation by
$$6$$
$$x_{1} = \frac{\pi n}{6} + \frac{5 \pi}{36}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{36}$$
$$x_{1} = \frac{\pi n}{6} + \frac{5 \pi}{36}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{36}$$
This roots
$$x_{1} = \frac{\pi n}{6} + \frac{5 \pi}{36}$$
$$x_{2} = \frac{\pi n}{6} - \frac{\pi}{36}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{6} + \frac{5 \pi}{36}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{6} - \frac{1}{10} + \frac{5 \pi}{36}$$
substitute to the expression
$$\cos{\left(6 x \right)} < - \frac{\sqrt{3}}{2}$$
$$\cos{\left(6 \left(\frac{\pi n}{6} - \frac{1}{10} + \frac{5 \pi}{36}\right) \right)} < - \frac{\sqrt{3}}{2}$$
___
/ 3 pi \ -\/ 3
-sin|- - + -- + pi*n| < -------
\ 5 3 / 2
but
___
/ 3 pi \ -\/ 3
-sin|- - + -- + pi*n| > -------
\ 5 3 / 2
Then
$$x < \frac{\pi n}{6} + \frac{5 \pi}{36}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{\pi n}{6} + \frac{5 \pi}{36} \wedge x < \frac{\pi n}{6} - \frac{\pi}{36}$$
_____
/ \
-------ο-------ο-------
x1 x2
Solving inequality on a graph
Rapid solution
[src]
/ / _____________\ / _____________\ \ | | / ___ | | / ___ | | | atan\\/ 7 + 4*\/ 3 / pi atan\\/ 7 + 4*\/ 3 / | And|x < - ---------------------- + --, ---------------------- < x| \ 3 3 3 /
$$x < - \frac{\operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}}{3} + \frac{\pi}{3} \wedge \frac{\operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}}{3} < x$$
(atan(sqrt(7 + 4*sqrt(3)))/3 < x)∧(x < -atan(sqrt(7 + 4*sqrt(3)))/3 + pi/3)
Rapid solution 2
[src]
/ _____________\ / _____________\
| / ___ | | / ___ |
atan\\/ 7 + 4*\/ 3 / atan\\/ 7 + 4*\/ 3 / pi
(----------------------, - ---------------------- + --)
3 3 3
$$x\ in\ \left(\frac{\operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}}{3}, - \frac{\operatorname{atan}{\left(\sqrt{4 \sqrt{3} + 7} \right)}}{3} + \frac{\pi}{3}\right)$$
x in Interval.open(atan(sqrt(4*sqrt(3) + 7))/3, -atan(sqrt(4*sqrt(3) + 7))/3 + pi/3)