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1+sinx^4-2sinx^2+a*sinx^2-2a=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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4 2 2 1 + sin (x) - 2*sin (x) + a*sin (x) - 2*a = 1
$$- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right) = 1$$
Detail solution
Given the equation
$$- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right) = 1$$
transform
$$- a \cos^{2}{\left(x \right)} - a + \cos^{4}{\left(x \right)} - 1 = 0$$
$$\left(- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right)\right) - 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$a w^{2} - 2 a + w^{4} - 2 w^{2} = 0$$
Do replacement
$$v = w^{2}$$
then the equation will be the:
$$- 2 a + v^{2} + v \left(a - 2\right) = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = a - 2$$
$$c = - 2 a$$
, then
The equation has two roots.
or
$$v_{1} = - \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1$$
$$v_{2} = - \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1$$
The final answer:
Because
$$v = w^{2}$$
then
$$w_{1} = \sqrt{v_{1}}$$
$$w_{2} = - \sqrt{v_{1}}$$
$$w_{3} = \sqrt{v_{2}}$$
$$w_{4} = - \sqrt{v_{2}}$$
then:
$$w_{1} = $$
$$\frac{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = \sqrt{- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
$$w_{2} = $$
$$\frac{\left(-1\right) \left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
$$w_{3} = $$
$$\frac{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = \sqrt{- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
$$w_{4} = $$
$$\frac{\left(-1\right) \left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} - 1 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} - 1 \right)} + \pi$$
$$- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right) = 1$$
transform
$$- a \cos^{2}{\left(x \right)} - a + \cos^{4}{\left(x \right)} - 1 = 0$$
$$\left(- 2 a + \left(a \sin^{2}{\left(x \right)} + \left(\left(\sin^{4}{\left(x \right)} + 1\right) - 2 \sin^{2}{\left(x \right)}\right)\right)\right) - 1 = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation:
$$a w^{2} - 2 a + w^{4} - 2 w^{2} = 0$$
Do replacement
$$v = w^{2}$$
then the equation will be the:
$$- 2 a + v^{2} + v \left(a - 2\right) = 0$$
This equation is of the form
a*v^2 + b*v + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$v_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$v_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = a - 2$$
$$c = - 2 a$$
, then
D = b^2 - 4 * a * c =
(-2 + a)^2 - 4 * (1) * (-2*a) = (-2 + a)^2 + 8*a
The equation has two roots.
v1 = (-b + sqrt(D)) / (2*a)
v2 = (-b - sqrt(D)) / (2*a)
or
$$v_{1} = - \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1$$
$$v_{2} = - \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1$$
The final answer:
Because
$$v = w^{2}$$
then
$$w_{1} = \sqrt{v_{1}}$$
$$w_{2} = - \sqrt{v_{1}}$$
$$w_{3} = \sqrt{v_{2}}$$
$$w_{4} = - \sqrt{v_{2}}$$
then:
$$w_{1} = $$
$$\frac{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = \sqrt{- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
$$w_{2} = $$
$$\frac{\left(-1\right) \left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
$$w_{3} = $$
$$\frac{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = \sqrt{- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
$$w_{4} = $$
$$\frac{\left(-1\right) \left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1\right)^{\frac{1}{2}}}{1} + \frac{0}{1} = - \sqrt{- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1}$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)}$$
$$x_{2} = 2 \pi n - \operatorname{asin}{\left(\frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} - 1 \right)}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{a}{2} - \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} + 1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \operatorname{asin}{\left(\frac{a}{2} + \frac{\sqrt{8 a + \left(a - 2\right)^{2}}}{2} - 1 \right)} + \pi$$
The graph
Rapid solution
[src]
/ / ____\\ / / ____\\ x1 = pi - re\asin\\/ -a // - I*im\asin\\/ -a //
$$x_{1} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi$$
/ / ____\\ / / ____\\ x2 = pi + I*im\asin\\/ -a // + re\asin\\/ -a //
$$x_{2} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi$$
/ / ____\\ / / ____\\ x3 = - re\asin\\/ -a // - I*im\asin\\/ -a //
$$x_{3} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}$$
/ / ____\\ / / ____\\ x4 = I*im\asin\\/ -a // + re\asin\\/ -a //
$$x_{4} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}$$
/ / ___\\ / / ___\\ x5 = pi - re\asin\\/ 2 // - I*im\asin\\/ 2 //
$$x_{5} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}$$
/ / ___\\ / / ___\\ x6 = pi + I*im\asin\\/ 2 // + re\asin\\/ 2 //
$$x_{6} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}$$
/ / ___\\ / / ___\\ x7 = - re\asin\\/ 2 // - I*im\asin\\/ 2 //
$$x_{7} = - \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}$$
/ / ___\\ / / ___\\ x8 = I*im\asin\\/ 2 // + re\asin\\/ 2 //
$$x_{8} = \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}$$
x8 = re(asin(sqrt(2))) + i*im(asin(sqrt(2)))
Sum and product of roots
[src]
sum
/ / ____\\ / / ____\\ / / ____\\ / / ____\\ / / ____\\ / / ____\\ / / ____\\ / / ____\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ / / ___\\ pi - re\asin\\/ -a // - I*im\asin\\/ -a // + pi + I*im\asin\\/ -a // + re\asin\\/ -a // + - re\asin\\/ -a // - I*im\asin\\/ -a // + I*im\asin\\/ -a // + re\asin\\/ -a // + pi - re\asin\\/ 2 // - I*im\asin\\/ 2 // + pi + I*im\asin\\/ 2 // + re\asin\\/ 2 // + - re\asin\\/ 2 // - I*im\asin\\/ 2 // + I*im\asin\\/ 2 // + re\asin\\/ 2 //
$$\left(\left(\left(\left(\left(\left(\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)\right) + \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)\right) + \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)$$
=
4*pi
$$4 \pi$$
product
/ / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ____\\ / / ____\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ \pi - re\asin\\/ -a // - I*im\asin\\/ -a ///*\pi + I*im\asin\\/ -a // + re\asin\\/ -a ///*\- re\asin\\/ -a // - I*im\asin\\/ -a ///*\I*im\asin\\/ -a // + re\asin\\/ -a ///*\pi - re\asin\\/ 2 // - I*im\asin\\/ 2 ///*\pi + I*im\asin\\/ 2 // + re\asin\\/ 2 ///*\- re\asin\\/ 2 // - I*im\asin\\/ 2 ///*\I*im\asin\\/ 2 // + re\asin\\/ 2 ///
$$\left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} - i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)$$
=
2 2 / / / ___\\ / / ___\\\ / / / ____\\ / / ____\\\ / / / ___\\ / / ___\\\ / / / ____\\ / / ____\\\ / / / ___\\ / / ___\\\ / / / ____\\ / / ____\\\ \I*im\asin\\/ 2 // + re\asin\\/ 2 /// *\I*im\asin\\/ -a // + re\asin\\/ -a /// *\pi + I*im\asin\\/ 2 // + re\asin\\/ 2 ///*\pi + I*im\asin\\/ -a // + re\asin\\/ -a ///*\-pi + I*im\asin\\/ 2 // + re\asin\\/ 2 ///*\-pi + I*im\asin\\/ -a // + re\asin\\/ -a ///
$$\left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right)^{2} \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)}\right)^{2} \left(- \pi + \operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} - \pi\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(\sqrt{- a} \right)}\right)} + \pi\right)$$
(i*im(asin(sqrt(2))) + re(asin(sqrt(2))))^2*(i*im(asin(sqrt(-a))) + re(asin(sqrt(-a))))^2*(pi + i*im(asin(sqrt(2))) + re(asin(sqrt(2))))*(pi + i*im(asin(sqrt(-a))) + re(asin(sqrt(-a))))*(-pi + i*im(asin(sqrt(2))) + re(asin(sqrt(2))))*(-pi + i*im(asin(sqrt(-a))) + re(asin(sqrt(-a))))