Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation tan(x)=1
-
Equation x^2-7x=0
-
Equation 27*x=1026
- Equation √3/(2x-11)=1/13
- Express {x} in terms of y where:
- -14*x+1*y=3
- 1*x+19*y=-10
- -16*x-7*y=-5
- -2*x+1*y=-8
- Integral of d{x}:
-
sin(3x)^2
- Derivative of:
- sin(3x)^2
-
Identical expressions
- sin(3x)^ two
- sinus of (3x) squared
- sinus of (3x) to the power of two
- sin(3x)2
- sin3x2
- sin(3x)²
- sin(3x) to the power of 2
- sin3x^2
sin(3x)^2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin^{2}{\left(3 x \right)} = 0$$
transform
$$\sin^{2}{\left(3 x \right)} = 0$$
$$\sin^{2}{\left(3 x \right)} + 0 = 0$$
Do replacement
$$w = \sin{\left(3 x \right)}$$
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
$$a = 1$$
$$b = 0$$
$$c = 0$$
, then
Because D = 0, then the equation has one root.
$$w_{1} = 0$$
do backward replacement
$$\sin{\left(3 x \right)} = w$$
Given the equation
$$\sin{\left(3 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$3 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
substitute w:
$$x_{1} = \frac{2 \pi n}{3} + \frac{\operatorname{asin}{\left(w_{1} \right)}}{3}$$
$$x_{1} = \frac{2 \pi n}{3} + \frac{\operatorname{asin}{\left(0 \right)}}{3}$$
$$x_{1} = \frac{2 \pi n}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{\operatorname{asin}{\left(w_{1} \right)}}{3} + \frac{\pi}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{\operatorname{asin}{\left(0 \right)}}{3} + \frac{\pi}{3}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
$$\sin^{2}{\left(3 x \right)} = 0$$
transform
$$\sin^{2}{\left(3 x \right)} = 0$$
$$\sin^{2}{\left(3 x \right)} + 0 = 0$$
Do replacement
$$w = \sin{\left(3 x \right)}$$
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
$$a = 1$$
$$b = 0$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (0) = 0
Because D = 0, then the equation has one root.
w = -b/2a = -0/2/(1)
$$w_{1} = 0$$
do backward replacement
$$\sin{\left(3 x \right)} = w$$
Given the equation
$$\sin{\left(3 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$3 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$3 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$3 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$3$$
substitute w:
$$x_{1} = \frac{2 \pi n}{3} + \frac{\operatorname{asin}{\left(w_{1} \right)}}{3}$$
$$x_{1} = \frac{2 \pi n}{3} + \frac{\operatorname{asin}{\left(0 \right)}}{3}$$
$$x_{1} = \frac{2 \pi n}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{\operatorname{asin}{\left(w_{1} \right)}}{3} + \frac{\pi}{3}$$
$$x_{2} = \frac{2 \pi n}{3} - \frac{\operatorname{asin}{\left(0 \right)}}{3} + \frac{\pi}{3}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
Sum and product of roots
[src]
sum
pi
0 + 0 + --
3
$$\left(0 + 0\right) + \frac{\pi}{3}$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
pi
1*0*--
3
$$1 \cdot 0 \frac{\pi}{3}$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = -70.1622358532669
x2 = -72.2566310094027
x3 = 83.7758041420923
x4 = 61.7846555856111
x5 = 43.9822971689772
x6 = -99.4837673664303
x7 = 63.8790506955489
x8 = 4.18879026129896
x9 = 59.6902602103008
x10 = -13.6135682324625
x11 = -50.2654824528405
x12 = -458.672527220906
x13 = -43.9822971750521
x14 = -55.5014702493934
x15 = 78.5398164276763
x16 = 30.3687289225719
x17 = 26.1799388472337
x18 = 96.3421746575137
x19 = 87.9645943339958
x20 = 48.1710874341111
x21 = -71.2094334160924
x22 = -21.9911485865686
x23 = 17.8023584422991
x24 = -11.5191731177358
x25 = -81.6814090352758
x26 = -87.9645943606119
x27 = 34.5575193908685
x28 = 19.896753540642
x29 = 72.2566310277355
x30 = -57.5958653860056
x31 = -2.09439503742649
x32 = -77.4926188103359
x33 = -35.6047168094363
x34 = 92.153384609279
x35 = 68.0678406435528
x36 = -83.7758040749281
x37 = -65.9734457660819
x38 = -24.0855436159154
x39 = 76.445421167341
x40 = -61.7846554917911
x41 = 74.351026078903
x42 = -15.7079632956456
x43 = 32.463124022232
x44 = -4.18879013064742
x45 = -48.1710872785046
x46 = -79.587013962129
x47 = 28.2743338654616
x48 = -37.6991118757886
x49 = 10.4719754514552
x50 = 21.9911485850825
x51 = 94.2477796093541
x52 = -26.1799387042774
x53 = 65.973445751897
x54 = 85.8701992723906
x55 = 54.4542725942726
x56 = -94.2477795706448
x57 = -28.2743339027943
x58 = -17.8023583237749
x59 = -39.7935069081029
x60 = 100.530964959912
x61 = -46.0766921948394
x62 = 37.699111685071
x63 = 70.1622360218137
x64 = 39.7935070144417
x65 = 98.4365697412614
x66 = -52.3598773534304
x67 = -59.6902604556648
x68 = 56.5486679087207
x69 = 0.0
x70 = -92.1533844285198
x71 = 41.8879021182991
x72 = 6.2831852847685
x73 = 12.566370486226
x74 = -33.5103216848946
x75 = 15.7079631676307
x76 = 83.7758041555675
x77 = -68.0678407742199
x78 = 50.2654824464452
x79 = 90.0589892608867
x80 = 8.37758034488241
x81 = 81.6814093004925
x82 = -6.28318536172932
x83 = -90.0589893540864
x84 = 52.3598775005845
x84 = 52.3598775005845
The graph