Mister Exam
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How to use it?
- Equation:
-
Equation cos(x/2)=(1/2)
-
Equation (x+9)^2=(x+6)^2
-
Equation (x^2-4)^2+(x^2-3*x-10)^2=0
-
Equation 11x-9=4x+19
- Express {x} in terms of y where:
- -6*x-17*y=14
- 18*x-12*y=-20
- -3*x-6*y=18
- 10*x+16*y=-6
-
Identical expressions
- six *sin(x)^(two)+ seven *cos(x)- seven = zero
- 6 multiply by sinus of (x) to the power of (2) plus 7 multiply by co sinus of e of (x) minus 7 equally 0
- six multiply by sinus of (x) to the power of (two) plus seven multiply by co sinus of e of (x) minus seven equally zero
- 6*sin(x)(2)+7*cos(x)-7=0
- 6*sinx2+7*cosx-7=0
- 6sin(x)^(2)+7cos(x)-7=0
- 6sin(x)(2)+7cos(x)-7=0
- 6sinx2+7cosx-7=0
- 6sinx^2+7cosx-7=0
- 6*sin(x)^(2)+7*cos(x)-7=O
-
Similar expressions
- 6*sin(x)^(2)+7*cos(x)+7=0
- 6*sin(x)^(2)-7*cos(x)-7=0
- 6*sinx^(2)+7*cosx-7=0
6*sin(x)^(2)+7*cos(x)-7=0 equation
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The solution
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[src]
2 6*sin (x) + 7*cos(x) - 7 = 0
$$6 \sin^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 7 = 0$$
Detail solution
Given the equation
$$6 \sin^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 7 = 0$$
transform
$$- 6 \cos^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 1 = 0$$
$$- 6 \cos^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 7 + 6 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -6$$
$$b = 7$$
$$c = -1$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = \frac{1}{6}$$
Simplify
$$w_{2} = 1$$
Simplify
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
$$6 \sin^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 7 = 0$$
transform
$$- 6 \cos^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 1 = 0$$
$$- 6 \cos^{2}{\left(x \right)} + 7 \cos{\left(x \right)} - 7 + 6 = 0$$
Do replacement
$$w = \cos{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -6$$
$$b = 7$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(7)^2 - 4 * (-6) * (-1) = 25
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = \frac{1}{6}$$
Simplify
$$w_{2} = 1$$
Simplify
do backward replacement
$$\cos{\left(x \right)} = w$$
Given the equation
$$\cos{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
Or
$$x = \pi n + \operatorname{acos}{\left(w \right)}$$
$$x = \pi n + \operatorname{acos}{\left(w \right)} - \pi$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{acos}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{1} = \pi n + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(w_{2} \right)}$$
$$x_{2} = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$x_{2} = \pi n$$
$$x_{3} = \pi n + \operatorname{acos}{\left(w_{1} \right)} - \pi$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{3} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{6} \right)}$$
$$x_{4} = \pi n + \operatorname{acos}{\left(w_{2} \right)} - \pi$$
$$x_{4} = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
$$x_{4} = \pi n - \pi$$
Sum and product of roots
[src]
sum
/ ____\ / ____\
|\/ 35 | |\/ 35 |
0 + 0 - 2*atan|------| + 2*atan|------|
\ 7 / \ 7 /
$$\left(- 2 \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)} + \left(0 + 0\right)\right) + 2 \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)}$$
=
0
$$0$$
product
/ ____\ / ____\
|\/ 35 | |\/ 35 |
1*0*-2*atan|------|*2*atan|------|
\ 7 / \ 7 /
$$1 \cdot 0 \left(- 2 \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)}\right) 2 \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)}$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
/ ____\
|\/ 35 |
x2 = -2*atan|------|
\ 7 /
$$x_{2} = - 2 \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)}$$
/ ____\
|\/ 35 |
x3 = 2*atan|------|
\ 7 /
$$x_{3} = 2 \operatorname{atan}{\left(\frac{\sqrt{35}}{7} \right)}$$
Numerical answer
[src]
x1 = 94.2477796093527
x2 = 75.3982237429669
x3 = 92.8444313601186
x4 = -99.1276166672982
x5 = 12.5663704669779
x6 = 11.163022366784
x7 = 26.5360894762936
x8 = -83.0847572409098
x9 = -100.530964774333
x10 = 50.2654824463592
x11 = -76.8015719337302
x12 = -25.132741248409
x13 = -69.115038339598
x14 = 32.8192747834731
x15 = 100.530964777434
x16 = -55.1453195170411
x17 = -20.252904169114
x18 = 81.6814091542942
x19 = 76.8015719337302
x20 = 13.9697188619344
x21 = 57.9520160121915
x22 = 23.7293929811431
x23 = -86.561246052939
x24 = 48.8621342098615
x25 = -64.2352013193711
x26 = 43.9822971693722
x27 = -32.8192747834731
x28 = -13.9697188619344
x29 = -7.68653355475479
x30 = -87.964594359001
x31 = 7.68653355475479
x32 = -57.9520160121915
x33 = 67.7116901314002
x34 = 70.5183866265507
x35 = 45.3856453978323
x36 = -51.6688307050119
x37 = -11.163022366784
x38 = -81.6814090376808
x39 = 37.6991120002774
x40 = 36.2957635955023
x41 = -43.9822971746367
x42 = -4.87983705960438
x43 = -56.5486672243813
x44 = -89.3679425480894
x45 = 30.0125782883227
x46 = 20.252904169114
x47 = -80.2780607457594
x48 = -50.2654823093403
x49 = -75.3982238414472
x50 = 89.3679425480894
x51 = 43.9822969538209
x52 = -26.5360894762936
x53 = -95.651127855269
x54 = -12.5663705035478
x55 = -23.7293929811431
x56 = -31.4159266875089
x57 = 86.561246052939
x58 = 31.4159266238823
x59 = 51.6688307050119
x60 = -30.0125782883227
x61 = 95.651127855269
x62 = -45.3856453978323
x63 = 39.1024600906527
x64 = -56.5486676368275
x65 = -67.7116901314002
x66 = 42.5789489026819
x67 = -36.2957635955023
x68 = -19633.5507366886
x69 = 56.5486676219708
x70 = 99.1276166672982
x71 = 61.4285048242207
x72 = 87.9645943355424
x73 = -1.40334824757521
x74 = 80.2780607457594
x75 = -37.6991118769879
x76 = -39.1024600906527
x77 = 6.28318528431365
x78 = -92.8444313601186
x79 = -6.28318515427505
x80 = 0.0
x81 = -42.5789489026819
x82 = -73.9948754385798
x83 = 62.8318529806628
x84 = 55.1453195170411
x85 = 64.2352013193711
x86 = 83.0847572409098
x87 = -94.2477794649012
x88 = 1.40334824757521
x89 = 4.87983705960438
x90 = 17.4462076739636
x91 = -48.8621342098615
x92 = 73.9948754385798
x93 = 18.8495558602598
x94 = -75.3982236624331
x94 = -75.3982236624331
The graph