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
- tg^ two *x/ two
- tg squared multiply by x divide by 2
- tg to the power of two multiply by x divide by two
- tg2*x/2
- tg²*x/2
- tg to the power of 2*x/2
- tg^2x/2
- tg2x/2
- tg^2*x divide by 2
tg^2*x/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\frac{\tan^{2}{\left(x \right)}}{2} = 0$$
transform
$$\frac{\tan^{2}{\left(x \right)}}{2} = 0$$
$$\frac{\tan^{2}{\left(x \right)}}{2} = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
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
$$a = \frac{1}{2}$$
$$b = 0$$
$$c = 0$$
, then
Because D = 0, then the equation has one root.
$$w_{1} = 0$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(0 \right)}$$
$$x_{1} = \pi n$$
$$\frac{\tan^{2}{\left(x \right)}}{2} = 0$$
transform
$$\frac{\tan^{2}{\left(x \right)}}{2} = 0$$
$$\frac{\tan^{2}{\left(x \right)}}{2} = 0$$
Do replacement
$$w = \tan{\left(x \right)}$$
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
$$a = \frac{1}{2}$$
$$b = 0$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1/2) * (0) = 0
Because D = 0, then the equation has one root.
w = -b/2a = -0/2/(1/2)
$$w_{1} = 0$$
do backward replacement
$$\tan{\left(x \right)} = w$$
Given the equation
$$\tan{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
Or
$$x = \pi n + \operatorname{atan}{\left(w \right)}$$
, where n - is a integer
substitute w:
$$x_{1} = \pi n + \operatorname{atan}{\left(w_{1} \right)}$$
$$x_{1} = \pi n + \operatorname{atan}{\left(0 \right)}$$
$$x_{1} = \pi n$$
Numerical answer
[src]
x1 = 59.6902602145004
x2 = 87.9645943363399
x3 = -43.9822971744223
x4 = 62.8318526257023
x5 = -59.6902604582742
x6 = -18.8495547465563
x7 = 91.1061883231058
x8 = -15.7079632968116
x9 = -3.14159313419367
x10 = -69.1150388967924
x11 = 78.53981615825
x12 = -87.9645943581507
x13 = 47.1238910903805
x14 = 75.3982242393431
x15 = -78.5398158757739
x16 = 31.4159270619219
x17 = 9.42477847373977
x18 = 56.5486675771117
x19 = 81.681409232902
x20 = -40.8407033559755
x21 = -56.5486672888531
x22 = -28.274333676669
x23 = 72.2566310277136
x24 = 43.9822971695754
x25 = 100.530964739312
x26 = 0.0
x27 = 6.28318528408307
x28 = 34.5575189958939
x29 = 84.8230012117849
x30 = 37.6991120687848
x31 = -97.3893724932976
x32 = -81.6814090388783
x33 = -84.8230005709274
x34 = 28.2743338651162
x35 = -21.9911485864129
x36 = -25.1327417214108
x37 = -40.8407056783072
x38 = 25.1327424765395
x39 = -84.82300290167
x40 = -94.2477794213743
x41 = 94.2477796093519
x42 = 15.7079634868755
x43 = 40.8407040393519
x44 = -62.8318542892494
x45 = 97.389372828611
x46 = -6.28318509494079
x47 = -53.4070753298489
x48 = -91.1061874849821
x49 = -72.2566308398808
x50 = -75.3982239115218
x51 = 59.690260650792
x52 = -100.530964462409
x53 = 12.5663704145927
x54 = 3.14159153945546
x55 = -65.973445764663
x56 = -62.8318519640761
x57 = 21.9911485852339
x58 = -18.8495570687636
x59 = -47.1238903089396
x60 = -50.265482258314
x61 = 25.1327401464195
x62 = -34.5575187016351
x63 = 50.2654824463153
x64 = 47.1238887521935
x65 = -31.4159267482748
x66 = -12.5663701141083
x67 = -37.6991118775909
x68 = 69.1150397058699
x69 = 18.8495554527235
x70 = 65.9734457532278
x71 = 53.4070756504516
x72 = 69.1150373568381
x73 = 91.1061859604104
x74 = -9.42477816679559
x75 = 3.14159386425559
x75 = 3.14159386425559