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-4)^2*(x-6)>0
- (x+4)^2*(x-2)*(x-3)<=0
- log(x-1)(5-x)<=0
- 15*2^x+7^(x-1)>2^(x+6)-7^x
- Derivative of:
-
sin(3*x+1)
- Graphing y =:
- sin(3*x+1)
-
Identical expressions
- sin(three *x+ one)> zero
- sinus of (3 multiply by x plus 1) greater than 0
- sinus of (three multiply by x plus one) greater than zero
- sin(3x+1)>0
- sin3x+1>0
-
Similar expressions
- -sin3x+1>=0
- sin(3*x-1)>0
- -sin(3*x)+1>=0
sin(3*x+1)>0 inequation
The solution
Detail solution
Given the inequality:
$$\sin{\left(3 x + 1 \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(3 x + 1 \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(3 x + 1 \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(3 x + 1 \right)} = 0$$
This equation is transformed to
$$3 x + 1 = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$3 x + 1 = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$3 x + 1 = 2 \pi n$$
$$3 x + 1 = 2 \pi n + \pi$$
, where n - is a integer
Move
$$1$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n - 1$$
$$3 x = 2 \pi n - 1 + \pi$$
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{2 \pi n}{3} - \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
$$x_{1} = \frac{2 \pi n}{3} - \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
This roots
$$x_{1} = \frac{2 \pi n}{3} - \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{2 \pi n}{3} - \frac{1}{3}\right) + - \frac{1}{10}$$
=
$$\frac{2 \pi n}{3} - \frac{13}{30}$$
substitute to the expression
$$\sin{\left(3 x + 1 \right)} > 0$$
$$\sin{\left(3 \left(\frac{2 \pi n}{3} - \frac{13}{30}\right) + 1 \right)} > 0$$
Then
$$x < \frac{2 \pi n}{3} - \frac{1}{3}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{2 \pi n}{3} - \frac{1}{3} \wedge x < \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
$$\sin{\left(3 x + 1 \right)} > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\sin{\left(3 x + 1 \right)} = 0$$
Solve:
Given the equation
$$\sin{\left(3 x + 1 \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\sin{\left(3 x + 1 \right)} = 0$$
This equation is transformed to
$$3 x + 1 = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$3 x + 1 = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$3 x + 1 = 2 \pi n$$
$$3 x + 1 = 2 \pi n + \pi$$
, where n - is a integer
Move
$$1$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n - 1$$
$$3 x = 2 \pi n - 1 + \pi$$
Divide both parts of the equation by
$$3$$
$$x_{1} = \frac{2 \pi n}{3} - \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
$$x_{1} = \frac{2 \pi n}{3} - \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
This roots
$$x_{1} = \frac{2 \pi n}{3} - \frac{1}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
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
$$x_{0} = x_{1} - \frac{1}{10}$$
=
$$\left(\frac{2 \pi n}{3} - \frac{1}{3}\right) + - \frac{1}{10}$$
=
$$\frac{2 \pi n}{3} - \frac{13}{30}$$
substitute to the expression
$$\sin{\left(3 x + 1 \right)} > 0$$
$$\sin{\left(3 \left(\frac{2 \pi n}{3} - \frac{13}{30}\right) + 1 \right)} > 0$$
sin(-3/10 + 2*pi*n) > 0
Then
$$x < \frac{2 \pi n}{3} - \frac{1}{3}$$
no execute
one of the solutions of our inequality is:
$$x > \frac{2 \pi n}{3} - \frac{1}{3} \wedge x < \frac{2 \pi n}{3} - \frac{1}{3} + \frac{\pi}{3}$$
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Solving inequality on a graph
Rapid solution
[src]
/ / / / ___ \ / _______________________\\\ / / / / ___ \\ / _______________________\\ \\ | | | |-sin(1/3) + \/ 3 *cos(1/3)| | / 2 2 ||| | 2*pi | | |\/ 3 *cos(1/3) + sin(1/3) || | / 2 2 || || Or|And|0 <= x, x < -I*|I*atan|--------------------------| + log\\/ cos (1/3) + sin (1/3) /||, And|x <= ----, -I*|I*|pi + atan|--------------------------|| + log\\/ cos (1/3) + sin (1/3) /| < x|| | | | | ___ | || | 3 | | | ___ || | || \ \ \ \\/ 3 *sin(1/3) + cos(1/3) / // \ \ \ \-cos(1/3) + \/ 3 *sin(1/3)// / //
$$\left(0 \leq x \wedge x < - i \left(\log{\left(\sqrt{\sin^{2}{\left(\frac{1}{3} \right)} + \cos^{2}{\left(\frac{1}{3} \right)}} \right)} + i \operatorname{atan}{\left(\frac{- \sin{\left(\frac{1}{3} \right)} + \sqrt{3} \cos{\left(\frac{1}{3} \right)}}{\sqrt{3} \sin{\left(\frac{1}{3} \right)} + \cos{\left(\frac{1}{3} \right)}} \right)}\right)\right) \vee \left(x \leq \frac{2 \pi}{3} \wedge - i \left(\log{\left(\sqrt{\sin^{2}{\left(\frac{1}{3} \right)} + \cos^{2}{\left(\frac{1}{3} \right)}} \right)} + i \left(\operatorname{atan}{\left(\frac{\sin{\left(\frac{1}{3} \right)} + \sqrt{3} \cos{\left(\frac{1}{3} \right)}}{- \cos{\left(\frac{1}{3} \right)} + \sqrt{3} \sin{\left(\frac{1}{3} \right)}} \right)} + \pi\right)\right) < x\right)$$
((0 <= x)∧(x < -i*(i*atan((-sin(1/3) + sqrt(3)*cos(1/3))/(sqrt(3)*sin(1/3) + cos(1/3))) + log(sqrt(cos(1/3)^2 + sin(1/3)^2)))))∨((x <= 2*pi/3)∧(-i*(i*(pi + atan((sqrt(3)*cos(1/3) + sin(1/3))/(-cos(1/3) + sqrt(3)*sin(1/3)))) + log(sqrt(cos(1/3)^2 + sin(1/3)^2))) < x))