Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation cos(x)-1=0
- Equation 64-x^2=0
-
Equation sin(5*x)=-1
-
Equation 7^x-2=49
- Express {x} in terms of y where:
- -17*x+9*y=-1
- -19*x-18*y=6
- -20*x+14*y=7
- -4*x-18*y=-8
-
Identical expressions
- two - one /ctg^2x= zero
- 2 minus 1 divide by ctg squared x equally 0
- two minus one divide by ctg squared x equally zero
- 2-1/ctg2x=0
- 2-1/ctg²x=0
- 2-1/ctg to the power of 2x=0
- 2-1/ctg^2x=O
- 2-1 divide by ctg^2x=0
-
Similar expressions
- 2+1/ctg^2x=0
2-1/ctg^2x=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$2 - \frac{1}{\cot^{2}{\left(x \right)}} = 0$$
transform
$$2 - \tan^{2}{\left(x \right)} = 0$$
$$2 - \frac{1}{\cot^{2}{\left(x \right)}} = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Given the equation
$$2 - \frac{1}{w^{2}} = 0$$
Because equation degree is equal to = -2 - contains the even number -2 in the numerator, then
the equation has two real roots.
Get the root -2-th degree of the equation sides:
We get:
$$\frac{1}{\sqrt{\frac{1}{w^{2}}}} = \frac{1}{\sqrt{2}}$$
$$\frac{1}{\sqrt{\frac{1}{w^{2}}}} = \frac{-1}{\sqrt{2}}$$
or
$$w = \frac{\sqrt{2}}{2}$$
$$w = - \frac{\sqrt{2}}{2}$$
Expand brackets in the right part
We get the answer: w = sqrt(2)/2
Expand brackets in the right part
We get the answer: w = -sqrt(2)/2
or
$$w_{1} = - \frac{\sqrt{2}}{2}$$
$$w_{2} = \frac{\sqrt{2}}{2}$$
The final answer:
$$w_{1} = - \frac{\sqrt{2}}{2}$$
$$w_{2} = \frac{\sqrt{2}}{2}$$
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
$$2 - \frac{1}{\cot^{2}{\left(x \right)}} = 0$$
transform
$$2 - \tan^{2}{\left(x \right)} = 0$$
$$2 - \frac{1}{\cot^{2}{\left(x \right)}} = 0$$
Do replacement
$$w = \cot{\left(x \right)}$$
Given the equation
$$2 - \frac{1}{w^{2}} = 0$$
Because equation degree is equal to = -2 - contains the even number -2 in the numerator, then
the equation has two real roots.
Get the root -2-th degree of the equation sides:
We get:
$$\frac{1}{\sqrt{\frac{1}{w^{2}}}} = \frac{1}{\sqrt{2}}$$
$$\frac{1}{\sqrt{\frac{1}{w^{2}}}} = \frac{-1}{\sqrt{2}}$$
or
$$w = \frac{\sqrt{2}}{2}$$
$$w = - \frac{\sqrt{2}}{2}$$
Expand brackets in the right part
w = sqrt2/2
We get the answer: w = sqrt(2)/2
Expand brackets in the right part
w = -sqrt2/2
We get the answer: w = -sqrt(2)/2
or
$$w_{1} = - \frac{\sqrt{2}}{2}$$
$$w_{2} = \frac{\sqrt{2}}{2}$$
The final answer:
$$w_{1} = - \frac{\sqrt{2}}{2}$$
$$w_{2} = \frac{\sqrt{2}}{2}$$
do backward replacement
$$\cot{\left(x \right)} = w$$
substitute w:
Rapid solution
[src]
/ ___\
|\/ 2 |
x1 = -acot|-----|
\ 2 /
$$x_{1} = - \operatorname{acot}{\left(\frac{\sqrt{2}}{2} \right)}$$
/ ___\
|\/ 2 |
x2 = acot|-----|
\ 2 /
$$x_{2} = \operatorname{acot}{\left(\frac{\sqrt{2}}{2} \right)}$$
x2 = acot(sqrt(2)/2)
Sum and product of roots
[src]
sum
/ ___\ / ___\
|\/ 2 | |\/ 2 |
- acot|-----| + acot|-----|
\ 2 / \ 2 /
$$- \operatorname{acot}{\left(\frac{\sqrt{2}}{2} \right)} + \operatorname{acot}{\left(\frac{\sqrt{2}}{2} \right)}$$
=
0
$$0$$
product
/ ___\ / ___\
|\/ 2 | |\/ 2 |
-acot|-----|*acot|-----|
\ 2 / \ 2 /
$$- \operatorname{acot}{\left(\frac{\sqrt{2}}{2} \right)} \operatorname{acot}{\left(\frac{\sqrt{2}}{2} \right)}$$
=
/ ___\
2|\/ 2 |
-acot |-----|
\ 2 /
$$- \operatorname{acot}^{2}{\left(\frac{\sqrt{2}}{2} \right)}$$
-acot(sqrt(2)/2)^2
Numerical answer
[src]
x1 = -2.18627603546528
x2 = -16.6632798860735
x3 = 32.3712431540224
x4 = 52.451758492902
x5 = -99.5756482967489
x6 = -39.8853878785428
x7 = 92.0615035722285
x8 = 96.4340556431591
x9 = 90.1508703359795
x10 = -11.6110539962347
x11 = 68.1597217608509
x12 = -52.451758492902
x13 = 55.5933511464918
x14 = 4.0969092717143
x15 = -48.0792064219714
x16 = -35.5128358076122
x17 = 11.6110539962347
x18 = -95405889.0850112
x19 = 19.8048725396633
x20 = -61.8765364536714
x21 = 74.4429070680305
x22 = 8.46946134264487
x23 = 48.0792064219714
x24 = -87.0092776823897
x25 = 26.0880578468429
x26 = -19.8048725396633
x27 = -63.7871696899204
x28 = -90.1508703359795
x29 = -26.0880578468429
x30 = 17.8942393034142
x31 = 80.7260923752101
x32 = 13.5216872324837
x33 = -13.5216872324837
x34 = 60.6455770363306
x35 = -46.1685731857224
x36 = 30.4606099177734
x37 = 41.7960211147918
x38 = -57.5039843827408
x39 = 76.3535403042795
x40 = -79.4951329578693
x41 = 38.654428461202
x42 = 39.8853878785428
x43 = 63.7871696899204
x44 = -1315.372005236
x45 = 10.3800945788939
x46 = 70.0703549971
x47 = -41.7960211147918
x48 = -85.7783182650489
x49 = -92.0615035722285
x50 = -96.4340556431591
x51 = -83.8676850287999
x52 = -68.1597217608509
x53 = -55.5933511464918
x54 = -10.3800945788939
x55 = 65.0181291072611
x56 = -30.4606099177734
x57 = -8.46946134264487
x58 = -24.1774246105938
x59 = -70.0703549971
x60 = -54.362391729151
x61 = -76.3535403042795
x62 = 61.8765364536714
x63 = -33.6022025713632
x64 = -77.5844997216203
x65 = 33.6022025713632
x66 = -4.0969092717143
x67 = 1940.54894330037
x68 = 83.8676850287999
x69 = 54.362391729151
x70 = -32.3712431540224
x71 = -27.3190172641836
x72 = 24.1774246105938
x73 = 46.1685731857224
x74 = 82.6367256114591
x75 = 98.3446888794081
x76 = -17.8942393034142
x77 = -74.4429070680305
x78 = 85.7783182650489
x79 = 2.18627603546528
x79 = 2.18627603546528