Mister Exam
tg(t)<1 inequation
The solution
Detail solution
Given the inequality:
$$\tan{\left(t \right)} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(t \right)} = 1$$
Solve:
Given the equation
$$\tan{\left(t \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$t = \pi n + \operatorname{atan}{\left(1 \right)}$$
Or
$$t = \pi n + \frac{\pi}{4}$$
, where n - is a integer
$$t_{1} = \pi n + \frac{\pi}{4}$$
$$t_{1} = \pi n + \frac{\pi}{4}$$
This roots
$$t_{1} = \pi n + \frac{\pi}{4}$$
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
$$t_{0} = t_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{4}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\tan{\left(t \right)} < 1$$
$$\tan{\left(\pi n - \frac{1}{10} + \frac{\pi}{4} \right)} < 1$$
the solution of our inequality is:
$$t < \pi n + \frac{\pi}{4}$$
$$\tan{\left(t \right)} < 1$$
To solve this inequality, we must first solve the corresponding equation:
$$\tan{\left(t \right)} = 1$$
Solve:
Given the equation
$$\tan{\left(t \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$t = \pi n + \operatorname{atan}{\left(1 \right)}$$
Or
$$t = \pi n + \frac{\pi}{4}$$
, where n - is a integer
$$t_{1} = \pi n + \frac{\pi}{4}$$
$$t_{1} = \pi n + \frac{\pi}{4}$$
This roots
$$t_{1} = \pi n + \frac{\pi}{4}$$
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
$$t_{0} = t_{1} - \frac{1}{10}$$
=
$$\left(\pi n + \frac{\pi}{4}\right) + - \frac{1}{10}$$
=
$$\pi n - \frac{1}{10} + \frac{\pi}{4}$$
substitute to the expression
$$\tan{\left(t \right)} < 1$$
$$\tan{\left(\pi n - \frac{1}{10} + \frac{\pi}{4} \right)} < 1$$
/ 1 pi \ tan|- -- + -- + pi*n| < 1 \ 10 4 /
the solution of our inequality is:
$$t < \pi n + \frac{\pi}{4}$$
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t1
Solving inequality on a graph
Rapid solution
[src]
/ / pi\ / pi \\ Or|And|0 <= t, t < --|, And|t <= pi, -- < t|| \ \ 4 / \ 2 //
$$\left(0 \leq t \wedge t < \frac{\pi}{4}\right) \vee \left(t \leq \pi \wedge \frac{\pi}{2} < t\right)$$
((0 <= t)∧(t < pi/4))∨((t <= pi)∧(pi/2 < t))
Rapid solution 2
[src]
pi pi
[0, --) U (--, pi]
4 2
$$t\ in\ \left[0, \frac{\pi}{4}\right) \cup \left(\frac{\pi}{2}, \pi\right]$$
t in Union(Interval.Ropen(0, pi/4), Interval.Lopen(pi/2, pi))