Mister Exam
Other calculators
-
How to use it?
- Equation:
- Equation 4b^2+17b-21=0
-
Equation tg(x)=sqrt3/2
- Equation (2sin((3pi)/2+2x)^2+sin(4x))/log3(2^(1/2)*sinx)=0
- Equation 9^x^2-x-5+6^x^2-x-4-180*4^x^2-X-7=0
- Express {x} in terms of y where:
- sqrt(x^2+y^2+4x-6y-12)+(x^2+10x-27)=8-|y-9|+|y-3|
- (x^2+y^2-1)^3+x^2*y^3=0
- -1*x-19*y=-10
- 4*x-16*y=6
-
Identical expressions
- (two sin((3pi)/ two + two x)^ two +sin(4x))/log3(2^(one /2)*sinx)= zero
- (2 sinus of ((3 Pi ) divide by 2 plus 2x) squared plus sinus of (4x)) divide by logarithm of 3(2 to the power of (1 divide by 2) multiply by sinus of x) equally 0
- (two sinus of ((3 Pi ) divide by two plus two x) to the power of two plus sinus of (4x)) divide by logarithm of 3(2 to the power of (one divide by 2) multiply by sinus of x) equally zero
- (2sin((3pi)/2+2x)2+sin(4x))/log3(2(1/2)*sinx)=0
- 2sin3pi/2+2x2+sin4x/log321/2*sinx=0
- (2sin((3pi)/2+2x)²+sin(4x))/log3(2^(1/2)*sinx)=0
- (2sin((3pi)/2+2x) to the power of 2+sin(4x))/log3(2 to the power of (1/2)*sinx)=0
- (2sin((3pi)/2+2x)^2+sin(4x))/log3(2^(1/2)sinx)=0
- (2sin((3pi)/2+2x)2+sin(4x))/log3(2(1/2)sinx)=0
- 2sin3pi/2+2x2+sin4x/log321/2sinx=0
- 2sin3pi/2+2x^2+sin4x/log32^1/2sinx=0
- (2sin((3pi)/2+2x)^2+sin(4x))/log3(2^(1/2)*sinx)=O
- (2sin((3pi) divide by 2+2x)^2+sin(4x)) divide by log3(2^(1 divide by 2)*sinx)=0
-
Similar expressions
- (2sin((3pi)/2+2x)^2-sin(4x))/log3(2^(1/2)*sinx)=0
- (2sin((3pi)/2-2x)^2+sin(4x))/log3(2^(1/2)*sinx)=0
(2sin((3pi)/2+2x)^2+sin(4x))/log3(2^(1/2)*sinx)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2/3*pi \
2*sin |---- + 2*x| + sin(4*x)
\ 2 /
----------------------------- = 0
/ / ___ \\
|log\\/ 2 *sin(x)/|
|-----------------|
\ log(3) /
$$\frac{\sin{\left(4 x \right)} + 2 \sin^{2}{\left(2 x + \frac{3 \pi}{2} \right)}}{\frac{1}{\log{\left(3 \right)}} \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}} = 0$$
Detail solution
Given the equation
$$\frac{\sin{\left(4 x \right)} + 2 \sin^{2}{\left(2 x + \frac{3 \pi}{2} \right)}}{\frac{1}{\log{\left(3 \right)}} \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}} = 0$$
transform
$$\frac{\left(\sqrt{2} \sin{\left(4 x + \frac{\pi}{4} \right)} + 1\right) \log{\left(3 \right)} - \log{\left(\sin{\left(x \right)} \right)} - \frac{\log{\left(2 \right)}}{2}}{\log{\left(\sin{\left(x \right)} \right)} + \frac{\log{\left(2 \right)}}{2}} = 0$$
$$-1 + \frac{\sin{\left(4 x \right)} + 2 \sin^{2}{\left(2 x + \frac{3 \pi}{2} \right)}}{\frac{1}{\log{\left(3 \right)}} \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}} = 0$$
Do replacement
$$w = \cos{\left(2 x \right)}$$
Given the equation:
$$-1 + \frac{\sin{\left(4 x \right)} + 2 \sin^{2}{\left(2 x + \frac{3 \pi}{2} \right)}}{\frac{1}{\log{\left(3 \right)}} \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}} = 0$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case
so we get the equation
$$\left(\sin{\left(4 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \log{\left(3 \right)} = \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}$$
$$\left(\sin{\left(4 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \log{\left(3 \right)} = \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}$$
Expand brackets in the left part
Expand brackets in the right part
This equation has no roots
do backward replacement
$$\cos{\left(2 x \right)} = w$$
Given the equation
$$\cos{\left(2 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
substitute w:
$$\frac{\sin{\left(4 x \right)} + 2 \sin^{2}{\left(2 x + \frac{3 \pi}{2} \right)}}{\frac{1}{\log{\left(3 \right)}} \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}} = 0$$
transform
$$\frac{\left(\sqrt{2} \sin{\left(4 x + \frac{\pi}{4} \right)} + 1\right) \log{\left(3 \right)} - \log{\left(\sin{\left(x \right)} \right)} - \frac{\log{\left(2 \right)}}{2}}{\log{\left(\sin{\left(x \right)} \right)} + \frac{\log{\left(2 \right)}}{2}} = 0$$
$$-1 + \frac{\sin{\left(4 x \right)} + 2 \sin^{2}{\left(2 x + \frac{3 \pi}{2} \right)}}{\frac{1}{\log{\left(3 \right)}} \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}} = 0$$
Do replacement
$$w = \cos{\left(2 x \right)}$$
Given the equation:
$$-1 + \frac{\sin{\left(4 x \right)} + 2 \sin^{2}{\left(2 x + \frac{3 \pi}{2} \right)}}{\frac{1}{\log{\left(3 \right)}} \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}} = 0$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case
a1 = (2*cos(2*x)^2 + sin(4*x))*log(3)
b1 = log(sqrt(2)*sin(x))
a2 = 1
b2 = 1
so we get the equation
$$\left(\sin{\left(4 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \log{\left(3 \right)} = \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}$$
$$\left(\sin{\left(4 x \right)} + 2 \cos^{2}{\left(2 x \right)}\right) \log{\left(3 \right)} = \log{\left(\sqrt{2} \sin{\left(x \right)} \right)}$$
Expand brackets in the left part
2*cos+2*x^2 + sin4*x)*log3 = log(sqrt(2)*sin(x))
Expand brackets in the right part
2*cos+2*x^2 + sin4*x)*log3 = logsqrt+2sinx)
This equation has no roots
do backward replacement
$$\cos{\left(2 x \right)} = w$$
Given the equation
$$\cos{\left(2 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$2 x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$2 x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
substitute w:
Rapid solution
[src]
-3*pi
x1 = -----
4
$$x_{1} = - \frac{3 \pi}{4}$$
-5*pi
x2 = -----
8
$$x_{2} = - \frac{5 \pi}{8}$$
-pi
x3 = ----
4
$$x_{3} = - \frac{\pi}{4}$$
-pi
x4 = ----
8
$$x_{4} = - \frac{\pi}{8}$$
3*pi
x5 = ----
8
$$x_{5} = \frac{3 \pi}{8}$$
7*pi
x6 = ----
8
$$x_{6} = \frac{7 \pi}{8}$$
x6 = 7*pi/8
Sum and product of roots
[src]
sum
3*pi 5*pi pi pi 3*pi 7*pi - ---- - ---- - -- - -- + ---- + ---- 4 8 4 8 8 8
$$\left(\left(\left(\left(- \frac{3 \pi}{4} - \frac{5 \pi}{8}\right) - \frac{\pi}{4}\right) - \frac{\pi}{8}\right) + \frac{3 \pi}{8}\right) + \frac{7 \pi}{8}$$
=
-pi ---- 2
$$- \frac{\pi}{2}$$
product
-3*pi -5*pi -pi -pi 3*pi 7*pi -----*-----*----*----*----*---- 4 8 4 8 8 8
$$\frac{7 \pi}{8} \frac{3 \pi}{8} \cdot - \frac{\pi}{8} \cdot - \frac{\pi}{4} \cdot - \frac{3 \pi}{4} \left(- \frac{5 \pi}{8}\right)$$
=
6 315*pi ------- 65536
$$\frac{315 \pi^{6}}{65536}$$
315*pi^6/65536
Numerical answer
[src]
x1 = 42.0188017417635
x2 = -31.8086256175967
x3 = 74.6128255227576
x4 = 29.0597320457056
x5 = -77.7544181763474
x6 = 34.164820107789
x7 = -21.2057504117311
x8 = 54.1924732744239
x9 = -47.5165888855456
x10 = -52.2289778659303
x11 = -69.9004365423729
x12 = 98.174770424681
x13 = 2.74889357189107
x14 = 43.5895980685584
x15 = 84.4303025652257
x16 = 90.7134878724053
x17 = 35.3429173528852
x18 = -63.6172512351933
x19 = -88.3572933822129
x20 = -44.7676953136546
x21 = -44.3749962319558
x22 = 4.31968989868597
x23 = 64.009950316892
x24 = 81.2887099116359
x25 = -75.7909227678538
x26 = -25.9181393921158
x27 = -91.4988860358027
x28 = -13.3517687777566
x29 = 13.7444678594553
x30 = -50.6581815391354
x31 = -9.8174770424681
x32 = -83.6449044018282
x33 = -3.53429173528852
x34 = -45.9457925587507
x35 = 10.2101761241668
x36 = -61.6537558266997
x37 = 0.0
x38 = -19.6349540849362
x39 = 68.329640215578
x40 = -6.67588438887831
x41 = -67.9369411338793
x42 = -121.736715326604
x43 = -30.2378292908018
x44 = -7.06858347057703
x45 = 46.7311907221482
x46 = 18.0641577581413
x47 = 30.6305283725005
x48 = -39.6626072515711
x49 = -16.1006623496477
x50 = 16.4933614313464
x51 = -41.233403578366
x52 = 60.4756585816035
x53 = 40.4480054149686
x54 = 49.872783375738
x55 = 79.717913584841
x56 = -38.484510006475
x57 = 3.92699081698724
x58 = 70.2931356240716
x59 = 5.89048622548086
x60 = 73.0420291959627
x61 = 20.0276531666349
x62 = -53.7997741927252
x63 = -27.4889357189107
x64 = 56.1559686829176
x65 = -97.7820713429823
x66 = -60.0829594999048
x67 = -1.96349540849362
x68 = 100.138265833175
x69 = 37.3064127613788
x70 = -23.9546439836222
x71 = -94.6404786893925
x72 = -71.4712328691678
x73 = 86.0010988920206
x74 = 78.1471172580461
x75 = -65.1880475619882
x76 = 66.7588438887831
x77 = 57.7267650097125
x78 = 12.1736715326604
x79 = -17.6714586764426
x80 = 93.8550805259951
x81 = -89.9280897090078
x82 = 62.0464549083984
x83 = -82.0741080750334
x84 = -74.2201264410589
x85 = -33.7721210260903
x86 = 35.7356164345839
x87 = 87.5718952188155
x88 = -38.0918109247762
x89 = -85.2157007286231
x90 = 24.3473430653209
x91 = -96.2112750161874
x92 = -8.24668071567321
x93 = 48.3019870489431
x94 = 26.3108384738145
x95 = -82.4668071567321
x95 = -82.4668071567321