Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- asin(x)<=п/2
- 7^x-5^(x-2)>2*7^(x-1)-118*5^(x-1)
- (2*x)^2-4*(x+2)>=0
- cos3x>√2/2
- Derivative of:
-
cos3x
- Graphing y =:
- cos3x
- Integral of d{x}:
-
cos3x
-
Identical expressions
- cos3x>√ two / two
- co sinus of e of 3x greater than √2 divide by 2
- co sinus of e of 3x greater than √ two divide by two
- cos3x>√2 divide by 2
-
Similar expressions
- cos(3*x+1)<=-(sqrt(2))/2
- sin(3*x)/2+cos(3*x)/2>0
- cos(3*x+1)<=-(√2/2)
cos3x>√2/2 inequation
The solution
You have entered
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\/ 2
cos(3*x) > -----
2
$$\cos{\left(3 x \right)} > \frac{\sqrt{2}}{2}$$
cos(3*x) > sqrt(2)/2
Detail solution
Given the inequality:
$$\cos{\left(3 x \right)} > \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\cos{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
Or
$$3 x = \pi n + \frac{\pi}{4}$$
$$3 x = \pi n - \frac{3 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{4}$$
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{4}$$
This roots
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{3} + \frac{\pi}{12}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{3} - \frac{1}{10} + \frac{\pi}{12}$$
substitute to the expression
$$\cos{\left(3 x \right)} > \frac{\sqrt{2}}{2}$$
$$\cos{\left(3 \left(\frac{\pi n}{3} - \frac{1}{10} + \frac{\pi}{12}\right) \right)} > \frac{\sqrt{2}}{2}$$
Then
$$x < \frac{\pi n}{3} + \frac{\pi}{12}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{\pi n}{3} + \frac{\pi}{12} \wedge x < \frac{\pi n}{3} - \frac{\pi}{4}$$
$$\cos{\left(3 x \right)} > \frac{\sqrt{2}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
Solve:
Given the equation
$$\cos{\left(3 x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = \pi n + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$3 x = \pi n - \pi + \operatorname{acos}{\left(\frac{\sqrt{2}}{2} \right)}$$
Or
$$3 x = \pi n + \frac{\pi}{4}$$
$$3 x = \pi n - \frac{3 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{4}$$
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{4}$$
This roots
$$x_{1} = \frac{\pi n}{3} + \frac{\pi}{12}$$
$$x_{2} = \frac{\pi n}{3} - \frac{\pi}{4}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{\pi n}{3} + \frac{\pi}{12}\right) + - \frac{1}{10}$$
=
$$\frac{\pi n}{3} - \frac{1}{10} + \frac{\pi}{12}$$
substitute to the expression
$$\cos{\left(3 x \right)} > \frac{\sqrt{2}}{2}$$
$$\cos{\left(3 \left(\frac{\pi n}{3} - \frac{1}{10} + \frac{\pi}{12}\right) \right)} > \frac{\sqrt{2}}{2}$$
___
/ 3 pi \ \/ 2
cos|- -- + -- + pi*n| > -----
\ 10 4 / 2
Then
$$x < \frac{\pi n}{3} + \frac{\pi}{12}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{\pi n}{3} + \frac{\pi}{12} \wedge x < \frac{\pi n}{3} - \frac{\pi}{4}$$
_____
/ \
-------ο-------ο-------
x1 x2
Solving inequality on a graph
Rapid solution
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/ / / _____________\\ / / _____________\ \\ | | | / ___ || | | / ___ | || | | 2*atan\\/ 3 - 2*\/ 2 /| | 2*pi 2*atan\\/ 3 - 2*\/ 2 / 2*pi || Or|And|0 <= x, x < ------------------------|, And|x <= ----, - ------------------------ + ---- < x|| \ \ 3 / \ 3 3 3 //
$$\left(0 \leq x \wedge x < \frac{2 \operatorname{atan}{\left(\sqrt{3 - 2 \sqrt{2}} \right)}}{3}\right) \vee \left(x \leq \frac{2 \pi}{3} \wedge - \frac{2 \operatorname{atan}{\left(\sqrt{3 - 2 \sqrt{2}} \right)}}{3} + \frac{2 \pi}{3} < x\right)$$
((0 <= x)∧(x < 2*atan(sqrt(3 - 2*sqrt(2)))/3))∨((x <= 2*pi/3)∧(-2*atan(sqrt(3 - 2*sqrt(2)))/3 + 2*pi/3 < x))
Rapid solution 2
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/ _____________\ / _____________\
| / ___ | | / ___ |
2*atan\\/ 3 - 2*\/ 2 / 2*atan\\/ 3 - 2*\/ 2 / 2*pi 2*pi
[0, ------------------------) U (- ------------------------ + ----, ----]
3 3 3 3
$$x\ in\ \left[0, \frac{2 \operatorname{atan}{\left(\sqrt{3 - 2 \sqrt{2}} \right)}}{3}\right) \cup \left(- \frac{2 \operatorname{atan}{\left(\sqrt{3 - 2 \sqrt{2}} \right)}}{3} + \frac{2 \pi}{3}, \frac{2 \pi}{3}\right]$$
x in Union(Interval.Ropen(0, 2*atan(sqrt(3 - 2*sqrt(2)))/3), Interval.Lopen(-2*atan(sqrt(3 - 2*sqrt(2)))/3 + 2*pi/3, 2*pi/3))