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Other calculators
- Square Inequality Step-by-Step
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-
How to use it?
- Inequation:
- y^2-4*x+8>0
- (x-4)^2>16-x^2
-
(|x+3|)+4*x>=6
- (sqrt(3)-(sqrt(13)))x>5/(sqrt(3)+sqrt(13))
-
Identical expressions
- cos(x/ five + two pi/ three)<-sqrt2/2
- co sinus of e of (x divide by 5 plus 2 Pi divide by 3) less than minus square root of 2 divide by 2
- co sinus of e of (x divide by five plus two Pi divide by three) less than minus square root of 2 divide by 2
- cos(x/5+2pi/3)<-√2/2
- cosx/5+2pi/3<-sqrt2/2
- cos(x divide by 5+2pi divide by 3)<-sqrt2 divide by 2
-
Similar expressions
- sin(x)>=(-sqrt(2))/2
- sin(t)<-sqrt(2)/2
- cos(x/5-2pi/3)<-sqrt2/2
- cos(x/5+2pi/3)<+sqrt2/2
cos(x/5+2pi/3)<-sqrt2/2 inequation
The solution
You have entered
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___ /x 2*pi\ -\/ 2 cos|- + ----| < ------- \5 3 / 2
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
cos(x/5 + (2*pi)/3) < (-sqrt(2))/2
Detail solution
Given the inequality:
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Solve:
Given the equation
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(\frac{x}{5} + \frac{\pi}{6} \right)} = \frac{\sqrt{2}}{2}$$
This equation is transformed to
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{4}$$
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{5} = 2 \pi n + \frac{\pi}{12}$$
$$\frac{x}{5} = 2 \pi n + \frac{7 \pi}{12}$$
Divide both parts of the equation by
$$\frac{1}{5}$$
$$x_{1} = 10 \pi n + \frac{5 \pi}{12}$$
$$x_{2} = 10 \pi n + \frac{35 \pi}{12}$$
$$x_{1} = 10 \pi n + \frac{5 \pi}{12}$$
$$x_{2} = 10 \pi n + \frac{35 \pi}{12}$$
This roots
$$x_{1} = 10 \pi n + \frac{5 \pi}{12}$$
$$x_{2} = 10 \pi n + \frac{35 \pi}{12}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(10 \pi n + \frac{5 \pi}{12}\right) + - \frac{1}{10}$$
=
$$10 \pi n - \frac{1}{10} + \frac{5 \pi}{12}$$
substitute to the expression
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\cos{\left(\frac{10 \pi n - \frac{1}{10} + \frac{5 \pi}{12}}{5} + \frac{2 \pi}{3} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
but
Then
$$x < 10 \pi n + \frac{5 \pi}{12}$$
no execute
one of the solutions of our inequality is:
$$x > 10 \pi n + \frac{5 \pi}{12} \wedge x < 10 \pi n + \frac{35 \pi}{12}$$
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
Solve:
Given the equation
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} = \frac{\left(-1\right) \sqrt{2}}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -1
The equation is transformed to
$$\sin{\left(\frac{x}{5} + \frac{\pi}{6} \right)} = \frac{\sqrt{2}}{2}$$
This equation is transformed to
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n + \frac{\pi}{4}$$
$$\frac{x}{5} + \frac{\pi}{6} = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{5} = 2 \pi n + \frac{\pi}{12}$$
$$\frac{x}{5} = 2 \pi n + \frac{7 \pi}{12}$$
Divide both parts of the equation by
$$\frac{1}{5}$$
$$x_{1} = 10 \pi n + \frac{5 \pi}{12}$$
$$x_{2} = 10 \pi n + \frac{35 \pi}{12}$$
$$x_{1} = 10 \pi n + \frac{5 \pi}{12}$$
$$x_{2} = 10 \pi n + \frac{35 \pi}{12}$$
This roots
$$x_{1} = 10 \pi n + \frac{5 \pi}{12}$$
$$x_{2} = 10 \pi n + \frac{35 \pi}{12}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(10 \pi n + \frac{5 \pi}{12}\right) + - \frac{1}{10}$$
=
$$10 \pi n - \frac{1}{10} + \frac{5 \pi}{12}$$
substitute to the expression
$$\cos{\left(\frac{x}{5} + \frac{2 \pi}{3} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
$$\cos{\left(\frac{10 \pi n - \frac{1}{10} + \frac{5 \pi}{12}}{5} + \frac{2 \pi}{3} \right)} < \frac{\left(-1\right) \sqrt{2}}{2}$$
___
/ 1 pi \ -\/ 2
-sin|- -- + -- + 2*pi*n| < -------
\ 50 4 / 2
but
___
/ 1 pi \ -\/ 2
-sin|- -- + -- + 2*pi*n| > -------
\ 50 4 / 2
Then
$$x < 10 \pi n + \frac{5 \pi}{12}$$
no execute
one of the solutions of our inequality is:
$$x > 10 \pi n + \frac{5 \pi}{12} \wedge x < 10 \pi n + \frac{35 \pi}{12}$$
_____
/ \
-------ο-------ο-------
x1 x2
Solving inequality on a graph
Rapid solution
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/ / _____________ \ / _____________ \ \ | | ___ ___ / ___ | | ___ ___ / ___ | | | | 3 - 2*\/ 3 \/ 2 *\/ 7 - 4*\/ 3 | | 3 - 2*\/ 3 \/ 2 *\/ 7 - 4*\/ 3 | | And|x < 10*atan|- --------------------------- + ---------------------------|, -10*atan|--------------------------- + ---------------------------| < x| | | ___ ___ ___ ___ ___ ___| | ___ ___ ___ ___ ___ ___| | \ \ 2 - \/ 3 - \/ 6 + 2*\/ 2 2 - \/ 3 - \/ 6 + 2*\/ 2 / \2 - \/ 3 - \/ 6 + 2*\/ 2 2 - \/ 3 - \/ 6 + 2*\/ 2 / /
$$x < 10 \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{7 - 4 \sqrt{3}}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} - \frac{3 - 2 \sqrt{3}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} \right)} \wedge - 10 \operatorname{atan}{\left(\frac{3 - 2 \sqrt{3}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} + \frac{\sqrt{2} \sqrt{7 - 4 \sqrt{3}}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} \right)} < x$$
(-10*atan((3 - 2*sqrt(3))/(2 - sqrt(3) - sqrt(6) + 2*sqrt(2)) + sqrt(2)*sqrt(7 - 4*sqrt(3))/(2 - sqrt(3) - sqrt(6) + 2*sqrt(2))) < x)∧(x < 10*atan(-(3 - 2*sqrt(3))/(2 - sqrt(3) - sqrt(6) + 2*sqrt(2)) + sqrt(2)*sqrt(7 - 4*sqrt(3))/(2 - sqrt(3) - sqrt(6) + 2*sqrt(2))))
Rapid solution 2
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/ _____________ \ / _____________ \
| ___ ___ / ___ | | ___ ___ / ___ |
| 3 - 2*\/ 3 \/ 2 *\/ 7 - 4*\/ 3 | | 3 - 2*\/ 3 \/ 2 *\/ 7 - 4*\/ 3 |
(-10*atan|--------------------------- + ---------------------------|, 10*atan|- --------------------------- + ---------------------------|)
| ___ ___ ___ ___ ___ ___| | ___ ___ ___ ___ ___ ___|
\2 - \/ 3 - \/ 6 + 2*\/ 2 2 - \/ 3 - \/ 6 + 2*\/ 2 / \ 2 - \/ 3 - \/ 6 + 2*\/ 2 2 - \/ 3 - \/ 6 + 2*\/ 2 /
$$x\ in\ \left(- 10 \operatorname{atan}{\left(\frac{3 - 2 \sqrt{3}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} + \frac{\sqrt{2} \sqrt{7 - 4 \sqrt{3}}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} \right)}, 10 \operatorname{atan}{\left(\frac{\sqrt{2} \sqrt{7 - 4 \sqrt{3}}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} - \frac{3 - 2 \sqrt{3}}{- \sqrt{6} - \sqrt{3} + 2 + 2 \sqrt{2}} \right)}\right)$$
x in Interval.open(-10*atan((3 - 2*sqrt(3))/(-sqrt(6) - sqrt(3) + 2 + 2*sqrt(2)) + sqrt(2)*sqrt(7 - 4*sqrt(3))/(-sqrt(6) - sqrt(3) + 2 + 2*sqrt(2))), 10*atan(sqrt(2)*sqrt(7 - 4*sqrt(3))/(-sqrt(6) - sqrt(3) + 2 + 2*sqrt(2)) - (3 - 2*sqrt(3))/(-sqrt(6) - sqrt(3) + 2 + 2*sqrt(2))))