Mister Exam
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How to use it?
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Equation 2^x=5
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Identical expressions
- 2sin(x+(pi/ three))+sqrt(three)= zero
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- 2sin(x+(pi divide by 3))+sqrt(3)=0
-
Similar expressions
- 2sin(x-(pi/3))+sqrt(3)=0
- 2sin(x+(pi/3))-sqrt(3)=0
2sin(x+(pi/3))+sqrt(3)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/ pi\ ___
2*sin|x + --| + \/ 3 = 0
\ 3 /
$$2 \sin{\left(x + \frac{\pi}{3} \right)} + \sqrt{3} = 0$$
Detail solution
Given the equation
$$2 \sin{\left(x + \frac{\pi}{3} \right)} + \sqrt{3} = 0$$
- this is the simplest trigonometric equation
We get:
$$2 \sin{\left(x + \frac{\pi}{3} \right)} = - \sqrt{3}$$
The equation is transformed to
$$\sin{\left(x + \frac{\pi}{3} \right)} = - \frac{\sqrt{3}}{2}$$
This equation is transformed to
$$x + \frac{\pi}{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$x + \frac{\pi}{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{3}$$
$$x + \frac{\pi}{3} = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n - \frac{2 \pi}{3}$$
$$x = 2 \pi n + \pi$$
$$2 \sin{\left(x + \frac{\pi}{3} \right)} + \sqrt{3} = 0$$
- this is the simplest trigonometric equation
Move sqrt(3) to right part of the equation
with the change of sign in sqrt(3)
We get:
$$2 \sin{\left(x + \frac{\pi}{3} \right)} = - \sqrt{3}$$
Divide both parts of the equation by 2
The equation is transformed to
$$\sin{\left(x + \frac{\pi}{3} \right)} = - \frac{\sqrt{3}}{2}$$
This equation is transformed to
$$x + \frac{\pi}{3} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)}$$
$$x + \frac{\pi}{3} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{3}}{2} \right)} + \pi$$
Or
$$x + \frac{\pi}{3} = 2 \pi n - \frac{\pi}{3}$$
$$x + \frac{\pi}{3} = 2 \pi n + \frac{4 \pi}{3}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$x = 2 \pi n - \frac{2 \pi}{3}$$
$$x = 2 \pi n + \pi$$
Sum and product of roots
[src]
sum
2*pi - ---- + pi 3
$$\pi - \frac{2 \pi}{3}$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
-2*pi -----*pi 3
$$- \frac{2 \pi}{3} \pi$$
=
2 -2*pi ------ 3
$$- \frac{2 \pi^{2}}{3}$$
-2*pi^2/3
Rapid solution
[src]
-2*pi
x1 = -----
3
$$x_{1} = - \frac{2 \pi}{3}$$
x2 = pi
$$x_{2} = \pi$$
x2 = pi
Numerical answer
[src]
x1 = 65.9734457253857
x2 = -91.106186954104
x3 = 4.18879020478639
x4 = -59.6902604182061
x5 = -21.9911485751286
x6 = -96.342174710087
x7 = -14.6607657167524
x8 = 21.9911485751286
x9 = 85.870199198121
x10 = 3.14159265358979
x11 = 29.3215314335047
x12 = -3.14159265358979
x13 = -15.707963267949
x14 = -53.4070751110265
x15 = -72.2566310325652
x16 = 84.8230016469244
x17 = 98.4365698124802
x18 = -64.9262481741891
x19 = -71.2094334813686
x20 = -46.0766922526503
x21 = 167.551608191456
x22 = -33.5103216382911
x23 = -58.6430628670095
x24 = 54.4542726622231
x25 = 73.3038285837618
x26 = -140.324471860344
x27 = -20.943951023932
x28 = 16.7551608191456
x29 = -90.0589894029074
x30 = 9.42477796076938
x31 = -65.9734457253857
x32 = -40.8407044966673
x33 = 97.3893722612836
x34 = 34.5575191894877
x35 = 35.6047167406843
x36 = 53.4070751110265
x37 = 92.1533845053006
x38 = -27.2271363311115
x39 = 67.0206432765823
x40 = 59.6902604182061
x41 = -28.2743338823081
x42 = 23.0383461263252
x43 = 60.7374579694027
x44 = 91.106186954104
x45 = 15.707963267949
x46 = 10.471975511966
x47 = -2.0943951023932
x48 = 41.8879020478639
x49 = -47.1238898038469
x50 = 72.2566310325652
x51 = -34.5575191894877
x52 = -83.7758040957278
x53 = 47.1238898038469
x54 = -97.3893722612836
x55 = 79.5870138909414
x56 = 40.8407044966673
x57 = -9.42477796076938
x58 = 78.5398163397448
x59 = -8.37758040957278
x60 = -78.5398163397448
x61 = -52.3598775598299
x62 = -39.7935069454707
x63 = -77.4926187885482
x64 = 28.2743338823081
x65 = 48.1710873550435
x66 = -84.8230016469244
x66 = -84.8230016469244