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- Equation:
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Equation sinx⋅tgx−(–√3)sinx=0
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Identical expressions
- sinx⋅tgx−(–√ three)sinx= zero
- sinus of x⋅tgx−(–√3) sinus of x equally 0
- sinus of x⋅tgx−(–√ three) sinus of x equally zero
- sinx⋅tgx−–√3sinx=0
- sinx⋅tgx−(–√3)sinx=O
sinx⋅tgx−(–√3)sinx=0 equation
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The solution
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[src]
___ sin(x)*tan(x) - -\/ 3 *sin(x) = 0
$$\sin{\left(x \right)} \tan{\left(x \right)} - - \sqrt{3} \sin{\left(x \right)} = 0$$
Detail solution
Given the equation:
$$\sin{\left(x \right)} \tan{\left(x \right)} - - \sqrt{3} \sin{\left(x \right)} = 0$$
Transform
$$\sin{\left(x \right)} \tan{\left(x \right)} - - \sqrt{3} \sin{\left(x \right)} = 0$$
$$\left(\tan{\left(x \right)} + \sqrt{3}\right) \sin{\left(x \right)} = 0$$
Consider each factor separately
$$\tan{\left(x \right)} + \sqrt{3} = 0$$
- 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:
$$\tan{\left(x \right)} = - \sqrt{3}$$
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
$$\sin{\left(x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x \right)} = 0$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x = 2 \pi n$$
$$x = 2 \pi n + \pi$$
, where n - is a integer
The final answer:
$$x_{1} = \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n + \pi$$
$$\sin{\left(x \right)} \tan{\left(x \right)} - - \sqrt{3} \sin{\left(x \right)} = 0$$
Transform
$$\sin{\left(x \right)} \tan{\left(x \right)} - - \sqrt{3} \sin{\left(x \right)} = 0$$
$$\left(\tan{\left(x \right)} + \sqrt{3}\right) \sin{\left(x \right)} = 0$$
Consider each factor separately
Step
$$\tan{\left(x \right)} + \sqrt{3} = 0$$
- 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:
$$\tan{\left(x \right)} = - \sqrt{3}$$
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
Step
$$\sin{\left(x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(x \right)} = 0$$
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$x = 2 \pi n$$
$$x = 2 \pi n + \pi$$
, where n - is a integer
The final answer:
$$x_{1} = \pi n - \frac{\pi}{3}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n + \pi$$
Sum and product of roots
[src]
sum
-pi
0 + ----
3
$$\left(0\right) + \left(- \frac{\pi}{3}\right)$$
=
-pi ---- 3
$$- \frac{\pi}{3}$$
product
-pi
0 * ----
3
$$\left(0\right) * \left(- \frac{\pi}{3}\right)$$
=
0
$$0$$
Numerical answer
[src]
x1 = 59.6902604182061
x2 = 47.1238898038469
x3 = 12.5663706143592
x4 = -59.6902604182061
x5 = -28.2743338823081
x6 = -12.5663706143592
x7 = 24.0855436775217
x8 = 6.28318530717959
x9 = -91.106186954104
x10 = -56.5486677646163
x11 = -25.1327412287183
x12 = -78.5398163397448
x13 = -31.4159265358979
x14 = -87.9645943005142
x15 = -43.9822971502571
x16 = 72.2566310325652
x17 = 75.398223686155
x18 = -51.3126800086333
x19 = 52.3598775598299
x20 = 78.5398163397448
x21 = 18.8495559215388
x22 = 0.0
x23 = -6.28318530717959
x24 = -94.2477796076938
x25 = -9.42477796076938
x26 = -62.8318530717959
x27 = -57.5958653158129
x28 = 40.8407044966673
x29 = 97.3893722612836
x30 = 14.6607657167524
x31 = -47.1238898038469
x32 = -18.8495559215388
x33 = -97.3893722612836
x34 = -65.9734457253857
x35 = -75.398223686155
x36 = 80.634211442138
x37 = 15.707963267949
x38 = -13.6135681655558
x39 = -29.3215314335047
x40 = -70.162235930172
x41 = -92.1533845053006
x42 = 3.14159265358979
x43 = -41.8879020478639
x44 = 53.4070751110265
x45 = -63.8790506229925
x46 = 43.9822971502571
x47 = 25.1327412287183
x48 = -7.33038285837618
x49 = 91.106186954104
x50 = 28.2743338823081
x51 = 30.3687289847013
x52 = -50.2654824574367
x53 = -72.2566310325652
x54 = 21.9911485751286
x55 = 56.5486677646163
x56 = -100.530964914873
x57 = -85.870199198121
x58 = 90.0589894029074
x59 = -40.8407044966673
x60 = -84.8230016469244
x61 = -53.4070751110265
x62 = -15.707963267949
x63 = 74.3510261349584
x64 = -81.6814089933346
x65 = -21.9911485751286
x66 = -69.1150383789755
x67 = 50.2654824574367
x68 = 9.42477796076938
x69 = 81.6814089933346
x70 = 84.8230016469244
x71 = -35.6047167406843
x72 = 68.0678408277789
x73 = 96.342174710087
x74 = -79.5870138909414
x75 = 36.6519142918809
x76 = -34.5575191894877
x77 = 31.4159265358979
x78 = 94.2477796076938
x79 = -37.6991118430775
x80 = 58.6430628670095
x81 = 62.8318530717959
x82 = 65.9734457253857
x83 = -19.8967534727354
x84 = 2.0943951023932
x85 = 8.37758040957278
x86 = 37.6991118430775
x87 = 34.5575191894877
x88 = 87.9645943005142
x89 = 69.1150383789755
x90 = 100.530964914873
x91 = 46.0766922526503
x92 = -3.14159265358979
x92 = -3.14159265358979
The graph