Mister Exam
sqrt(3)sinx-3cosx=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
___ \/ 3 *sin(x) - 3*cos(x) = 0
$$\sqrt{3} \sin{\left(x \right)} - 3 \cos{\left(x \right)} = 0$$
Detail solution
Given the equation
$$\sqrt{3} \sin{\left(x \right)} - 3 \cos{\left(x \right)} = 0$$
transform:
$$\frac{\sqrt{3} \sin{\left(x \right)}}{\cos{\left(x \right)}} = 3$$
or
$$\sqrt{3} \tan{\left(x \right)} = 3$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\tan{\left(x \right)} = - \sqrt{3}$$
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(- \sqrt{3} \right)}$$
Or
$$x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
$$\sqrt{3} \sin{\left(x \right)} - 3 \cos{\left(x \right)} = 0$$
transform:
$$\frac{\sqrt{3} \sin{\left(x \right)}}{\cos{\left(x \right)}} = 3$$
or
$$\sqrt{3} \tan{\left(x \right)} = 3$$
- this is the simplest trigonometric equation
Divide both parts of the equation by sqrt(3)
The equation is transformed to
$$\tan{\left(x \right)} = - \sqrt{3}$$
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(- \sqrt{3} \right)}$$
Or
$$x = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Sum and product of roots
[src]
sum
pi
0 + --
3
$$0 + \frac{\pi}{3}$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
pi 1*-- 3
$$1 \frac{\pi}{3}$$
=
pi -- 3
$$\frac{\pi}{3}$$
pi/3
Numerical answer
[src]
x1 = 104.71975511966
x2 = 67.0206432765823
x3 = 10.471975511966
x4 = 60.7374579694027
x5 = 98.4365698124802
x6 = -93.2005820564972
x7 = 26.1799387799149
x8 = -55.5014702134197
x9 = -99.4837673636768
x10 = 70.162235930172
x11 = -11.5191730631626
x12 = 19.8967534727354
x13 = -46.0766922526503
x14 = -27.2271363311115
x15 = -80.634211442138
x16 = 7.33038285837618
x17 = -71.2094334813686
x18 = 13.6135681655558
x19 = 73.3038285837618
x20 = -52.3598775598299
x21 = -96.342174710087
x22 = 16.7551608191456
x23 = -42.9350995990605
x24 = 82.7286065445312
x25 = -36.6519142918809
x26 = 35.6047167406843
x27 = -77.4926187885482
x28 = -17.8023583703422
x29 = 32.4631240870945
x30 = -2.0943951023932
x31 = -454.483737219323
x32 = 51.3126800086333
x33 = -61.7846555205993
x34 = -8.37758040957278
x35 = 1.0471975511966
x36 = 38.7463093942741
x37 = 41.8879020478639
x38 = -5.23598775598299
x39 = 76.4454212373516
x40 = 29.3215314335047
x41 = 63.8790506229925
x42 = -30.3687289847013
x43 = -58.6430628670095
x44 = -24.0855436775217
x45 = 79.5870138909414
x46 = 54.4542726622231
x47 = 45.0294947014537
x48 = -39.7935069454707
x49 = 89.0117918517108
x50 = -74.3510261349584
x51 = -68.0678408277789
x52 = -90.0589894029074
x53 = 85.870199198121
x54 = -33.5103216382911
x55 = 92.1533845053006
x56 = -20.943951023932
x57 = -64.9262481741891
x58 = -49.2182849062401
x59 = 57.5958653158129
x60 = 95.2949771588904
x61 = 23.0383461263252
x62 = -86.9173967493176
x63 = 4.18879020478639
x64 = -14.6607657167524
x65 = -83.7758040957278
x66 = 48.1710873550435
x66 = 48.1710873550435