Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
-
How to use it?
- Inequation:
- √(x^2-2x)>4-x
- 3^(x+2)>81
- 3^x>7*2^x
- (x+2)/5>=2*x/4
- Sum of series:
- sin
- Integral of d{x}:
- sin
- Derivative of:
-
sin
-
Identical expressions
- sin<=-(√ three)/ two
- sinus of less than or equal to minus (√3) divide by 2
- sinus of less than or equal to minus (√ three) divide by two
- sin<=-√3/2
- sin<=-(√3) divide by 2
-
Similar expressions
- sin<=+(√3)/2
sin<=-(√3)/2 inequation
The solution
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[src]
___
-\/ 3
sin(x) <= -------
2
$$\sin{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{2}$$
sin(x) <= (-sqrt(3))/2
Detail solution
Given the inequality:
$$\sin{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$x = 2 \pi n - \frac{\pi}{3}$$
$$x = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
$$x_{1} = 2 \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{1} = 2 \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \frac{4 \pi}{3}$$
This roots
$$x_{1} = 2 \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \frac{4 \pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \frac{\pi}{3}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{\pi}{3} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{2}$$
$$\sin{\left(2 \pi n - \frac{\pi}{3} - \frac{1}{10} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{2}$$
one of the solutions of our inequality is:
$$x \leq 2 \pi n - \frac{\pi}{3}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 2 \pi n - \frac{\pi}{3}$$
$$x \geq 2 \pi n + \frac{4 \pi}{3}$$
$$\sin{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
Solve:
Given the equation
$$\sin{\left(x \right)} = \frac{\left(-1\right) \sqrt{3}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$x = 2 \pi n - \frac{\pi}{3}$$
$$x = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
$$x_{1} = 2 \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{1} = 2 \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \frac{4 \pi}{3}$$
This roots
$$x_{1} = 2 \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n + \frac{4 \pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(2 \pi n - \frac{\pi}{3}\right) + - \frac{1}{10}$$
=
$$2 \pi n - \frac{\pi}{3} - \frac{1}{10}$$
substitute to the expression
$$\sin{\left(x \right)} \leq \frac{\left(-1\right) \sqrt{3}}{2}$$
$$\sin{\left(2 \pi n - \frac{\pi}{3} - \frac{1}{10} \right)} \leq \frac{\left(-1\right) \sqrt{3}}{2}$$
___
/1 pi \ -\/ 3
-sin|-- + -- - 2*pi*n| <= -------
\10 3 / 2
one of the solutions of our inequality is:
$$x \leq 2 \pi n - \frac{\pi}{3}$$
_____ _____
\ /
-------•-------•-------
x1 x2Other solutions will get with the changeover to the next point
etc.
The answer:
$$x \leq 2 \pi n - \frac{\pi}{3}$$
$$x \geq 2 \pi n + \frac{4 \pi}{3}$$
Solving inequality on a graph
Rapid solution
[src]
/4*pi 5*pi\ And|---- <= x, x <= ----| \ 3 3 /
$$\frac{4 \pi}{3} \leq x \wedge x \leq \frac{5 \pi}{3}$$
(4*pi/3 <= x)∧(x <= 5*pi/3)
Rapid solution 2
[src]
4*pi 5*pi [----, ----] 3 3
$$x\ in\ \left[\frac{4 \pi}{3}, \frac{5 \pi}{3}\right]$$
x in Interval(4*pi/3, 5*pi/3)