Mister Exam
Other calculators
- Square Inequality Step-by-Step
- Inequality with the module Step by Step
- Logarithmic inequality online
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How to use it?
- Inequation:
- (5-x)/3<2x-3/2
- (x+7)/2>(8−x)/3
- ((2-x^2)(x-3)^2)/((x+1)(x^2-3x-4))>=0
- (x+1)*(x-3)*(x+9)/((x-7)*(x-7))<0
- Integral of d{x}:
-
cost
- Derivative of:
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cost
-
Identical expressions
- cost>− zero . five
- co sinus of e of t greater than −0.5
- co sinus of e of t greater than − zero . five
cost>−0.5 inequation
The solution
Detail solution
Given the inequality:
$$\cos{\left(t \right)} > - \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(t \right)} = - \frac{1}{2}$$
Solve:
Given the equation
$$\cos{\left(t \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$t = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$t = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$t = \pi n + \frac{2 \pi}{3}$$
$$t = \pi n - \frac{\pi}{3}$$
, where n - is a integer
$$t_{1} = \pi n + \frac{2 \pi}{3}$$
$$t_{2} = \pi n - \frac{\pi}{3}$$
$$t_{1} = \pi n + \frac{2 \pi}{3}$$
$$t_{2} = \pi n - \frac{\pi}{3}$$
This roots
$$t_{1} = \pi n + \frac{2 \pi}{3}$$
$$t_{2} = \pi n - \frac{\pi}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$t_{0} < t_{1}$$
For example, let's take the point
$$t_{0} = t_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{2 \pi}{3}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{2 \pi}{3}$$
substitute to the expression
$$\cos{\left(t \right)} > - \frac{1}{2}$$
$$\cos{\left(\pi n - \frac{1}{10} + \frac{2 \pi}{3} \right)} > - \frac{1}{2}$$
one of the solutions of our inequality is:
$$t < \pi n + \frac{2 \pi}{3}$$
Other solutions will get with the changeover to the next point
etc.
The answer:
$$t < \pi n + \frac{2 \pi}{3}$$
$$t > \pi n - \frac{\pi}{3}$$
$$\cos{\left(t \right)} > - \frac{1}{2}$$
To solve this inequality, we must first solve the corresponding equation:
$$\cos{\left(t \right)} = - \frac{1}{2}$$
Solve:
Given the equation
$$\cos{\left(t \right)} = - \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$t = \pi n + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
$$t = \pi n - \pi + \operatorname{acos}{\left(- \frac{1}{2} \right)}$$
Or
$$t = \pi n + \frac{2 \pi}{3}$$
$$t = \pi n - \frac{\pi}{3}$$
, where n - is a integer
$$t_{1} = \pi n + \frac{2 \pi}{3}$$
$$t_{2} = \pi n - \frac{\pi}{3}$$
$$t_{1} = \pi n + \frac{2 \pi}{3}$$
$$t_{2} = \pi n - \frac{\pi}{3}$$
This roots
$$t_{1} = \pi n + \frac{2 \pi}{3}$$
$$t_{2} = \pi n - \frac{\pi}{3}$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$t_{0} < t_{1}$$
For example, let's take the point
$$t_{0} = t_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{2 \pi}{3}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{2 \pi}{3}$$
substitute to the expression
$$\cos{\left(t \right)} > - \frac{1}{2}$$
$$\cos{\left(\pi n - \frac{1}{10} + \frac{2 \pi}{3} \right)} > - \frac{1}{2}$$
/ 1 pi \
-sin|- -- + -- + pi*n| > -1/2
\ 10 6 / one of the solutions of our inequality is:
$$t < \pi n + \frac{2 \pi}{3}$$
_____ _____
\ /
-------ο-------ο-------
t1 t2Other solutions will get with the changeover to the next point
etc.
The answer:
$$t < \pi n + \frac{2 \pi}{3}$$
$$t > \pi n - \frac{\pi}{3}$$
Solving inequality on a graph
Rapid solution 2
[src]
2*pi 4*pi
[0, ----) U (----, 2*pi]
3 3
$$t\ in\ \left[0, \frac{2 \pi}{3}\right) \cup \left(\frac{4 \pi}{3}, 2 \pi\right]$$
t in Union(Interval.Ropen(0, 2*pi/3), Interval.Lopen(4*pi/3, 2*pi))
Rapid solution
[src]
/ / 2*pi\ / 4*pi \\ Or|And|0 <= t, t < ----|, And|t <= 2*pi, ---- < t|| \ \ 3 / \ 3 //
$$\left(0 \leq t \wedge t < \frac{2 \pi}{3}\right) \vee \left(t \leq 2 \pi \wedge \frac{4 \pi}{3} < t\right)$$
((0 <= t)∧(t < 2*pi/3))∨((t <= 2*pi)∧(4*pi/3 < t))