Mister Exam
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How to use it?
- Equation:
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Equation 6/(x-2)+5/x=3
-
Equation sin(x)=4
-
Equation 9^x-8*3^x-9=0
-
Equation x^2-6x-27=0
- Express {x} in terms of y where:
- 16*x-17*y=-15
- -4*x-4*y=19
- 15*x+1*y=19
- -16*x-10*y=4
- Integral of d{x}:
-
sin^2(x)cos^2(x)
- Graphing y =:
- sin^2(x)cos^2(x)
-
Identical expressions
- sin^ two (x)cos^ two (x)
- sinus of squared (x) co sinus of e of squared (x)
- sinus of to the power of two (x) co sinus of e of to the power of two (x)
- sin2(x)cos2(x)
- sin2xcos2x
- sin²(x)cos²(x)
- sin to the power of 2(x)cos to the power of 2(x)
- sin^2xcos^2x
sin^2(x)cos^2(x) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 2 sin (x)*cos (x) = 0
$$\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = 0$$
transform
$$\frac{1}{8} - \frac{\cos{\left(4 x \right)}}{8} = 0$$
$$\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = \sin^{2}{\left(x \right)}$$
$$b = 0$$
$$c = 0$$
, then
Because D = 0, then the equation has one root.
$$w_{1} = 0$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
$$\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = 0$$
transform
$$\frac{1}{8} - \frac{\cos{\left(4 x \right)}}{8} = 0$$
$$\sin^{2}{\left(x \right)} \cos^{2}{\left(x \right)} = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
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
$$a = \sin^{2}{\left(x \right)}$$
$$b = 0$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (sin(x)^2) * (0) = 0
Because D = 0, then the equation has one root.
w = -b/2a = -0/2/(sin(x)^2)
$$w_{1} = 0$$
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(0 \right)}$$
$$x_{1} = \pi n + \frac{\pi}{2}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{2} = \pi n - \pi + \operatorname{acos}{\left(0 \right)}$$
$$x_{2} = \pi n - \frac{\pi}{2}$$
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
-pi
x2 = ----
2
$$x_{2} = - \frac{\pi}{2}$$
pi
x3 = --
2
$$x_{3} = \frac{\pi}{2}$$
x3 = pi/2
Sum and product of roots
[src]
sum
pi pi - -- + -- 2 2
$$- \frac{\pi}{2} + \frac{\pi}{2}$$
=
0
$$0$$
product
-pi pi 0*----*-- 2 2
$$\frac{\pi}{2} \cdot 0 \left(- \frac{\pi}{2}\right)$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = -7.85398150696156
x2 = -17.2787595621355
x3 = -86.3937978789102
x4 = -39.2699081045218
x5 = -61.2610566398387
x6 = 34.5575190717885
x7 = 21.9911485851564
x8 = 92.6769832182628
x9 = 67.5442420634706
x10 = -89.5353907315491
x11 = 95.8185760424586
x12 = -45.5530935761698
x13 = -28.2743337586152
x14 = -94.247779486083
x15 = 100.530964798296
x16 = -31.4159266517141
x17 = -75.3982237985682
x18 = 89.5353906153414
x19 = 37.6991119665793
x20 = -72.2566309100272
x21 = 64.4026493150839
x22 = 0.0
x23 = 23.5619449483644
x24 = 65.9734457525462
x25 = 15.7079633917898
x26 = -42.4115007432387
x27 = -53.4070752253874
x28 = 81.6814091152362
x29 = 117.809724442492
x30 = -95.818575868455
x31 = -1.57079626356835
x32 = 72.2566310277248
x33 = 4.71238898608896
x34 = -1.57079642013166
x35 = -81.6814090370675
x36 = 42.4115007365289
x37 = -51.8362786915081
x38 = 6.28318528443138
x39 = 48.6946860958663
x40 = -6.28318518328035
x41 = 26.7035375390573
x42 = -65.9734457653935
x43 = -59.6902604569585
x44 = 78.5398162225044
x45 = -9.42477807759933
x46 = -29.8451301000724
x47 = 86.3937978937855
x48 = 7.85398173011892
x49 = 12.5663704969137
x50 = -80.1106125854791
x51 = 59.690260541069
x52 = -37.6991118766796
x53 = -20.4203521774723
x54 = -83.2522051669813
x55 = -58.1194645366003
x56 = -50.2654823342013
x57 = -87.9645943594276
x58 = 45.5530935075531
x59 = 50.2654824463816
x60 = -97.3893725907902
x61 = -58.1194640062544
x62 = 94.247779609353
x63 = -64.402649310466
x64 = -73.8274272808521
x65 = 28.2743338652921
x66 = 73.8274274646672
x67 = 87.9645943351391
x68 = -23.5619449982306
x69 = 20.4203521581227
x70 = -43.9822971747455
x71 = -67.5442421539445
x72 = 29.8451303084991
x73 = -100.530965206253
x74 = -97.3893723711949
x75 = -36.128315427252
x76 = 51.8362788866811
x77 = 95.8185756842062
x78 = 70.6858346557926
x79 = -21.9911485864927
x80 = 36.1283154718409
x81 = 56.5486676469942
x82 = 43.9822971692691
x83 = 80.1106131511482
x84 = -15.7079632962205
x85 = 70.6858338406532
x86 = 1.57079638652515
x87 = -14.1371668484631
x88 = 72.256630710694
x89 = 14.1371670778185
x89 = 14.1371670778185