Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2+2x+1=0
-
Equation x*7=56
-
Equation x^2=-5
-
Equation x*x=x
- Express {x} in terms of y where:
- -4*x+18*y=-2
- -2*x+2*y=-7
- -16*x+2*y=20
- -10*x-15*y=-2
- Derivative of:
-
sin(x)^(3)
-
Identical expressions
- sin(x)^(seven)=sin(x)^(three)
- sinus of (x) to the power of (7) equally sinus of (x) to the power of (3)
- sinus of (x) to the power of (seven) equally sinus of (x) to the power of (three)
- sin(x)(7)=sin(x)(3)
- sinx7=sinx3
- sinx^7=sinx^3
-
Similar expressions
- sinx^(7)=sinx^(3)
sin(x)^(7)=sin(x)^(3) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
7 3 sin (x) = sin (x)
$$\sin^{7}{\left(x \right)} = \sin^{3}{\left(x \right)}$$
Detail solution
Given the equation
$$\sin^{7}{\left(x \right)} = \sin^{3}{\left(x \right)}$$
transform
$$\sin^{7}{\left(x \right)} - \sin^{3}{\left(x \right)} = 0$$
$$\sin^{7}{\left(x \right)} - \sin^{3}{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation
$$w^{7} - w^{3} = 0$$
Obviously:
next,
transform
$$w^{4} = 1$$
Because equation degree is equal to = 4 - contains the even number 4 in the numerator, then
the equation has two real roots.
Get the root 4-th degree of the equation sides:
We get:
$$\sqrt[4]{w^{4}} = \sqrt[4]{1}$$
$$\sqrt[4]{w^{4}} = \left(-1\right) \sqrt[4]{1}$$
or
$$w = 1$$
$$w = -1$$
We get the answer: w = 1
We get the answer: w = -1
or
$$w_{1} = -1$$
$$w_{2} = 1$$
All other 2 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$z^{4} = 1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = 1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{4 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = 1$$
so
$$\cos{\left(4 p \right)} = 1$$
and
$$\sin{\left(4 p \right)} = 0$$
then
$$p = \frac{\pi N}{2}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{3} = - i$$
$$z_{4} = i$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
$$w_{1} = -1$$
$$w_{2} = 1$$
$$w_{3} = - i$$
$$w_{4} = i$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
$$\sin^{7}{\left(x \right)} = \sin^{3}{\left(x \right)}$$
transform
$$\sin^{7}{\left(x \right)} - \sin^{3}{\left(x \right)} = 0$$
$$\sin^{7}{\left(x \right)} - \sin^{3}{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
Given the equation
$$w^{7} - w^{3} = 0$$
Obviously:
w0 = 0
next,
transform
$$w^{4} = 1$$
Because equation degree is equal to = 4 - contains the even number 4 in the numerator, then
the equation has two real roots.
Get the root 4-th degree of the equation sides:
We get:
$$\sqrt[4]{w^{4}} = \sqrt[4]{1}$$
$$\sqrt[4]{w^{4}} = \left(-1\right) \sqrt[4]{1}$$
or
$$w = 1$$
$$w = -1$$
We get the answer: w = 1
We get the answer: w = -1
or
$$w_{1} = -1$$
$$w_{2} = 1$$
All other 2 root(s) is the complex numbers.
do replacement:
$$z = w$$
then the equation will be the:
$$z^{4} = 1$$
Any complex number can presented so:
$$z = r e^{i p}$$
substitute to the equation
$$r^{4} e^{4 i p} = 1$$
where
$$r = 1$$
- the magnitude of the complex number
Substitute r:
$$e^{4 i p} = 1$$
Using Euler’s formula, we find roots for p
$$i \sin{\left(4 p \right)} + \cos{\left(4 p \right)} = 1$$
so
$$\cos{\left(4 p \right)} = 1$$
and
$$\sin{\left(4 p \right)} = 0$$
then
$$p = \frac{\pi N}{2}$$
where N=0,1,2,3,...
Looping through the values of N and substituting p into the formula for z
Consequently, the solution will be for z:
$$z_{1} = -1$$
$$z_{2} = 1$$
$$z_{3} = - i$$
$$z_{4} = i$$
do backward replacement
$$z = w$$
$$w = z$$
The final answer:
w0 = 0
$$w_{1} = -1$$
$$w_{2} = 1$$
$$w_{3} = - i$$
$$w_{4} = i$$
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:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$x_{1} = 2 \pi n + \frac{\pi}{2}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(-1 \right)}$$
$$x_{2} = 2 \pi n - \frac{\pi}{2}$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
$$x_{3} = 2 \pi n + \frac{\pi}{2}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(-1 \right)} + \pi$$
$$x_{4} = 2 \pi n + \frac{3 \pi}{2}$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
-pi
x2 = ----
2
$$x_{2} = - \frac{\pi}{2}$$
pi
x3 = --
2
$$x_{3} = \frac{\pi}{2}$$
x4 = pi
$$x_{4} = \pi$$
3*pi
x5 = ----
2
$$x_{5} = \frac{3 \pi}{2}$$
/ ___\ x6 = -I*log\1 + \/ 2 /
$$x_{6} = - i \log{\left(1 + \sqrt{2} \right)}$$
/ ___\ x7 = I*log\1 + \/ 2 /
$$x_{7} = i \log{\left(1 + \sqrt{2} \right)}$$
/ ___\ x8 = pi - I*log\1 + \/ 2 /
$$x_{8} = \pi - i \log{\left(1 + \sqrt{2} \right)}$$
/ ___\ x9 = pi + I*log\1 + \/ 2 /
$$x_{9} = \pi + i \log{\left(1 + \sqrt{2} \right)}$$
x9 = pi + i*log(1 + sqrt(2))
Sum and product of roots
[src]
sum
pi pi 3*pi / ___\ / ___\ / ___\ / ___\ - -- + -- + pi + ---- - I*log\1 + \/ 2 / + I*log\1 + \/ 2 / + pi - I*log\1 + \/ 2 / + pi + I*log\1 + \/ 2 / 2 2 2
$$\left(\left(\pi - i \log{\left(1 + \sqrt{2} \right)}\right) + \left(\left(\left(\left(\left(- \frac{\pi}{2} + \frac{\pi}{2}\right) + \pi\right) + \frac{3 \pi}{2}\right) - i \log{\left(1 + \sqrt{2} \right)}\right) + i \log{\left(1 + \sqrt{2} \right)}\right)\right) + \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$
=
9*pi ---- 2
$$\frac{9 \pi}{2}$$
product
-pi pi 3*pi / / ___\\ / ___\ / / ___\\ / / ___\\ 0*----*--*pi*----*\-I*log\1 + \/ 2 //*I*log\1 + \/ 2 /*\pi - I*log\1 + \/ 2 //*\pi + I*log\1 + \/ 2 // 2 2 2
$$i \log{\left(1 + \sqrt{2} \right)} - i \log{\left(1 + \sqrt{2} \right)} \frac{3 \pi}{2} \pi \frac{\pi}{2} \cdot 0 \left(- \frac{\pi}{2}\right) \left(\pi - i \log{\left(1 + \sqrt{2} \right)}\right) \left(\pi + i \log{\left(1 + \sqrt{2} \right)}\right)$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 31.4160149957498
x2 = -25.1328215415017
x3 = -94.2477148387717
x4 = 12.5663016330717
x5 = -29.8451301040263
x6 = 9.42486773034122
x7 = -23.5619449858515
x8 = 75.398308669411
x9 = 53.4071619904715
x10 = 15.7080359478585
x11 = 73.8274274492725
x12 = 50.2654784091763
x13 = 56.548601802997
x14 = 56.5486865602073
x15 = -91.1062562648809
x16 = -51.8362786933717
x17 = -86.3937981701709
x18 = -14.1371668579582
x19 = 86.3937978998509
x20 = 80.1106128166908
x21 = 84.8229129491484
x22 = 7.8539817193106
x23 = 18.849471166143
x24 = 14.137167042574
x25 = -97.3894482129513
x26 = 62.8317653966523
x27 = 65.9733528515637
x28 = -69.1151120925136
x29 = -89.5353907133972
x30 = -56.548700133589
x31 = -37.6991397180627
x32 = -12.5662797695825
x33 = -45.5530935620733
x34 = 6.28313660475366
x35 = -50.2654144093295
x36 = -80.1106125916069
x37 = 84.8230486056857
x38 = -15.7079741487691
x39 = 95.8185760252944
x40 = 94.2477801894819
x41 = -31.4159984095489
x42 = -95.8185758688041
x43 = -87.9645855232631
x44 = -58.1194640134133
x45 = 6.28317668316519
x46 = -7.85398151622223
x47 = 64.4026493221587
x48 = 51.8362788729298
x49 = -72.2565646051863
x50 = -62.8318866935603
x51 = 65.9733867426446
x52 = -6.28311413230237
x53 = 87.9646128709672
x54 = 37.699081465691
x55 = 28.2742981408013
x56 = -1.57079640929584
x57 = -37.6991249572634
x58 = -9.42484840473099
x59 = -36.128315435532
x60 = -65.9734547031488
x61 = 78.5397519330028
x62 = 29.8451302962741
x63 = -67.5442421379331
x64 = -87.9646059635419
x65 = -59.6902757413449
x66 = -34.557427474802
x67 = -21.9911516404123
x68 = -73.8274272817137
x69 = -53.4071483801154
x70 = -47.1239671384874
x71 = -15.7079501160313
x72 = 97.3894549759772
x73 = 72.2566292957557
x74 = 102.101760895016
x75 = -100.530871604917
x76 = 81.6814859467927
x77 = 21.9911516417501
x78 = 87.9646063082987
x79 = 0.0
x80 = -28.2742642515502
x81 = -78.5397234102849
x82 = 28.274327536783
x83 = 37.699185988278
x84 = 43.9823032525727
x85 = -81.6814265005678
x86 = 42.4115007447272
x87 = -75.3982983150702
x88 = -43.9823032311159
x89 = 34.5574517026801
x90 = -43.9823054715965
x91 = 40.8406181206108
x92 = -56.5485753615375
x93 = 65.9734548120817
x94 = 59.6903359882051
x95 = -3.14167541100922
x96 = 20.4203521675702
x97 = 100.530902091752
x97 = 100.530902091752