Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
-
sin(x-pi/3)>=1/2
- x^2*log512(4-x)≥log2(x^2-8x+16)
- x^2-34*x+91<(-3)*x^2+18*x-69
- log0.5x>=1
- Graphing y =:
- sin(x-pi/3)
- Integral of d{x}:
-
sin(x-pi/3)
-
Identical expressions
- sin(x-pi/ three)>= one / two
- sinus of (x minus Pi divide by 3) greater than or equal to 1 divide by 2
- sinus of (x minus Pi divide by three) greater than or equal to one divide by two
- sinx-pi/3>=1/2
- sin(x-pi divide by 3)>=1 divide by 2
-
Similar expressions
- (cos(x)+pi/2)*(sin(x)-pi/3)*(tg^2(x)-1/3)>=0
- 2*sin((x-pi)/3)<3
- sin(x+pi/3)>=1/2
-
What you mean?
- sin((x - pi)/3) >= 1/2
- sin(x - pi/3) >= 1/2
- sin(x - pi/3) >= 1/2
sin(x-pi/3)>=1/2 inequation
The solution
You have entered
[src]
/ pi\ sin|x - --| >= 1/2 \ 3 /
$$\sin{\left(x - \frac{\pi}{3} \right)} \geq \frac{1}{2}$$
sin(x - pi/3) >= 1/2
Detail solution
Given the inequality:
$$\sin{\left(x - \frac{\pi}{3} \right)} \geq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x - \frac{\pi}{3} \right)} = \frac{1}{2}$$
Solve:
Given the equation
$$\sin{\left(x - \frac{\pi}{3} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by $-1$
The equation is transformed to
$$\cos{\left(x + \frac{\pi}{6} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x + \frac{\pi}{6} = 2 \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x + \frac{\pi}{6} = 2 \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$x + \frac{\pi}{6} = 2 \pi n + \frac{2 \pi}{3}$$
$$x + \frac{\pi}{6} = 2 \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation with the opposite sign, in total:
$$x = 2 \pi n + \frac{\pi}{2}$$
$$x = 2 \pi n - \frac{\pi}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{2}\right) - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{2}$$
substitute to the expression
$$\sin{\left(x - \frac{\pi}{3} \right)} \geq \frac{1}{2}$$
$$\sin{\left(\left(2 \pi n - \frac{1}{10} + \frac{\pi}{2}\right) - \frac{\pi}{3} \right)} \geq \frac{1}{2}$$
but
Then
$$x \leq 2 \pi n + \frac{\pi}{2}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n + \frac{\pi}{2} \wedge x \leq 2 \pi n - \frac{\pi}{2}$$
$$\sin{\left(x - \frac{\pi}{3} \right)} \geq \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x - \frac{\pi}{3} \right)} = \frac{1}{2}$$
Solve:
Given the equation
$$\sin{\left(x - \frac{\pi}{3} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by $-1$
The equation is transformed to
$$\cos{\left(x + \frac{\pi}{6} \right)} = - \frac{1}{2}$$
This equation is transformed to
$$x + \frac{\pi}{6} = 2 \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$x + \frac{\pi}{6} = 2 \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$x + \frac{\pi}{6} = 2 \pi n + \frac{2 \pi}{3}$$
$$x + \frac{\pi}{6} = 2 \pi n - \frac{\pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{6}$$
to right part of the equation with the opposite sign, in total:
$$x = 2 \pi n + \frac{\pi}{2}$$
$$x = 2 \pi n - \frac{\pi}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
This roots
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} \leq x_{1}$$
For example, let's take the point
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n + \frac{\pi}{2}\right) - \frac{1}{10}$$
=
$$2 \pi n - \frac{1}{10} + \frac{\pi}{2}$$
substitute to the expression
$$\sin{\left(x - \frac{\pi}{3} \right)} \geq \frac{1}{2}$$
$$\sin{\left(\left(2 \pi n - \frac{1}{10} + \frac{\pi}{2}\right) - \frac{\pi}{3} \right)} \geq \frac{1}{2}$$
/1 pi\ cos|-- + --| >= 1/2 \10 3 /
but
/1 pi\ cos|-- + --| < 1/2 \10 3 /
Then
$$x \leq 2 \pi n + \frac{\pi}{2}$$
no execute
one of the solutions of our inequality is:
$$x \geq 2 \pi n + \frac{\pi}{2} \wedge x \leq 2 \pi n - \frac{\pi}{2}$$
_____
/ \
-------•-------•-------
x_1 x_2
Solving inequality on a graph
Rapid solution
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/ / / _____________________________________________________________________________________________________________________________\\ \ | | / / / -pi*I \ \\ | / / -pi*I \ / -pi*I \ / -pi*I \ / -pi*I \ || | | | | | | _________ ------| || | / | _________ ------| | _________ ------| | _________ ------| | _________ ------| || | | | | | ___ | / 3 ____ 3 | || | / | / 3 ____ 3 | 2| / 3 ____ 3 | 2| / 3 ____ 3 | ___ | / 3 ____ 3 | || | | | | | 1 + 2*\/ 3 *im\\/ -\/ -1 *e / || | / 1 3*re\\/ -\/ -1 *e / 3*im \\/ -\/ -1 *e / 3*re \\/ -\/ -1 *e / \/ 3 *im\\/ -\/ -1 *e / || | And|x <= -I*|I*|pi + atan|------------------------------------------|| + log| / - - -------------------------- + --------------------------- + --------------------------- + ------------------------------ ||, 0 < x| | | | | / -pi*I \|| \\/ 4 4 4 4 4 /| | | | | | | _________ ------||| | | | | | | ___ ___ | / 3 ____ 3 ||| | | \ \ \ \- \/ 3 + 2*\/ 3 *re\\/ -\/ -1 *e /// / /
$$x \leq - i \left(\log{\left(\sqrt{\frac{\sqrt{3} \operatorname{im}{\left(\sqrt{- \sqrt[3]{-1}} e^{- \frac{i \pi}{3}}\right)}}{4} + \frac{3 \left(\operatorname{re}{\left(\sqrt{- \sqrt[3]{-1}} e^{- \frac{i \pi}{3}}\right)}\right)^{2}}{4} + \frac{1}{4} - \frac{3 \operatorname{re}{\left(\sqrt{- \sqrt[3]{-1}} e^{- \frac{i \pi}{3}}\right)}}{4} + \frac{3 \left(\operatorname{im}{\left(\sqrt{- \sqrt[3]{-1}} e^{- \frac{i \pi}{3}}\right)}\right)^{2}}{4}} \right)} + i \left(\operatorname{atan}{\left(\frac{2 \sqrt{3} \operatorname{im}{\left(\sqrt{- \sqrt[3]{-1}} e^{- \frac{i \pi}{3}}\right)} + 1}{- \sqrt{3} + 2 \sqrt{3} \operatorname{re}{\left(\sqrt{- \sqrt[3]{-1}} e^{- \frac{i \pi}{3}}\right)}} \right)} + \pi\right)\right) \wedge 0 < x$$
(0 < x)∧(x <= -i*(i*(pi + atan((1 + 2*sqrt(3)*im(sqrt(-(-1)^(1/3))*exp(-pi*i/3)))/(-sqrt(3) + 2*sqrt(3)*re(sqrt(-(-1)^(1/3))*exp(-pi*i/3))))) + log(sqrt(1/4 - 3*re(sqrt(-(-1)^(1/3))*exp(-pi*i/3))/4 + 3*im(sqrt(-(-1)^(1/3))*exp(-pi*i/3))^2/4 + 3*re(sqrt(-(-1)^(1/3))*exp(-pi*i/3))^2/4 + sqrt(3)*im(sqrt(-(-1)^(1/3))*exp(-pi*i/3))/4))))
The graph