Mister Exam
8sin^2x/2+3sinx-4>0 inequation
The solution
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2
8*sin (x)
--------- + 3*sin(x) - 4 > 0
2
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 > 0$$
3*sin(x) + (8*sin(x)^2)/2 - 4 > 0
Detail solution
Given the inequality:
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 = 0$$
Solve:
Given the equation
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 = 0$$
transform
$$3 \sin{\left(x \right)} - 4 \cos^{2}{\left(x \right)} = 0$$
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = 3$$
$$c = -4$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = - \frac{3}{8} + \frac{\sqrt{73}}{8}$$
$$w_{2} = - \frac{\sqrt{73}}{8} - \frac{3}{8}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{73}}{8} - \frac{3}{8} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{3}{8} + \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{73}}{8} - \frac{3}{8} \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = \pi + \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{2} = \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{3} = - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{4} = - \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
Exclude the complex solutions:
$$x_{1} = \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{2} = - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
This roots
$$x_{2} = - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
=
$$- \frac{1}{10} - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
substitute to the expression
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 > 0$$
Then
$$x < - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} \wedge x < \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 > 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 = 0$$
Solve:
Given the equation
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 = 0$$
transform
$$3 \sin{\left(x \right)} - 4 \cos^{2}{\left(x \right)} = 0$$
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 4$$
$$b = 3$$
$$c = -4$$
, then
D = b^2 - 4 * a * c =
(3)^2 - 4 * (4) * (-4) = 73
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = - \frac{3}{8} + \frac{\sqrt{73}}{8}$$
$$w_{2} = - \frac{\sqrt{73}}{8} - \frac{3}{8}$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{73}}{8} - \frac{3}{8} \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{3}{8} + \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{3} = 2 \pi n + \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi - \operatorname{asin}{\left(- \frac{\sqrt{73}}{8} - \frac{3}{8} \right)}$$
$$x_{4} = 2 \pi n + \pi + \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = \pi + \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
$$x_{2} = \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{3} = - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{4} = - \operatorname{asin}{\left(\frac{3}{8} + \frac{\sqrt{73}}{8} \right)}$$
Exclude the complex solutions:
$$x_{1} = \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
$$x_{2} = - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
This roots
$$x_{2} = - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
$$x_{1} = \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
is the points with change the sign of the inequality expression.
First define with the sign to the leftmost point:
$$x_{0} < x_{2}$$
For example, let's take the point
$$x_{0} = x_{2} - \frac{1}{10}$$
=
$$- \frac{1}{10} - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
=
$$- \frac{1}{10} - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
substitute to the expression
$$\left(3 \sin{\left(x \right)} + \frac{8 \sin^{2}{\left(x \right)}}{2}\right) - 4 > 0$$
/ / ____\\
2| 1 |3 \/ 73 ||
8*sin |- -- - asin|- - ------|| / / ____\\
\ 10 \8 8 // | 1 |3 \/ 73 ||
------------------------------- + 3*sin|- -- - asin|- - ------|| - 4 > 0
1 \ 10 \8 8 //
2 / / ____\\ / / ____\\
|1 |3 \/ 73 || 2|1 |3 \/ 73 ||
-4 - 3*sin|-- + asin|- - ------|| + 4*sin |-- + asin|- - ------|| > 0
\10 \8 8 // \10 \8 8 //
Then
$$x < - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)}$$
no execute
one of the solutions of our inequality is:
$$x > - \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} \wedge x < \operatorname{asin}{\left(\frac{3}{8} - \frac{\sqrt{73}}{8} \right)} + \pi$$
_____
/ \
-------ο-------ο-------
x2 x1
Solving inequality on a graph
Rapid solution
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/ / _____________\ / _____________\ \ | | ___ / ____ | | ___ / ____ | | | |\/ 6 *\/ -3 + \/ 73 | |\/ 6 *\/ -3 + \/ 73 | | And|x < pi - atan|----------------------|, atan|----------------------| < x| \ \ 6 / \ 6 / /
$$x < \pi - \operatorname{atan}{\left(\frac{\sqrt{6} \sqrt{-3 + \sqrt{73}}}{6} \right)} \wedge \operatorname{atan}{\left(\frac{\sqrt{6} \sqrt{-3 + \sqrt{73}}}{6} \right)} < x$$
(atan(sqrt(6)*sqrt(-3 + sqrt(73))/6) < x)∧(x < pi - atan(sqrt(6)*sqrt(-3 + sqrt(73))/6))
Rapid solution 2
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/ _____________\ / _____________\
| ___ / ____ | | ___ / ____ |
|\/ 6 *\/ -3 + \/ 73 | |\/ 6 *\/ -3 + \/ 73 |
(atan|----------------------|, pi - atan|----------------------|)
\ 6 / \ 6 /
$$x\ in\ \left(\operatorname{atan}{\left(\frac{\sqrt{6} \sqrt{-3 + \sqrt{73}}}{6} \right)}, \pi - \operatorname{atan}{\left(\frac{\sqrt{6} \sqrt{-3 + \sqrt{73}}}{6} \right)}\right)$$
x in Interval.open(atan(sqrt(6)*sqrt(-3 + sqrt(73))/6), pi - atan(sqrt(6)*sqrt(-3 + sqrt(73))/6))