Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation sin(x)=1
-
Equation k^2+6*k+13=0
-
Equation log(27)*x=1/3
-
Equation 2*x^4-15*x^3+35*x^2-30x+8=0
- Express {x} in terms of y where:
- 4*x+13*y=13
- -16*x+14*y=-9
- -14*x+19*y=-3
- -11*x+10*y=-9
-
Identical expressions
- one - two sin^2(2x)= zero
- 1 minus 2 sinus of squared (2x) equally 0
- one minus two sinus of squared (2x) equally zero
- 1-2sin2(2x)=0
- 1-2sin22x=0
- 1-2sin²(2x)=0
- 1-2sin to the power of 2(2x)=0
- 1-2sin^22x=0
- 1-2sin^2(2x)=O
-
Similar expressions
- 1+2sin^2(2x)=0
1-2sin^2(2x)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$1 - 2 \sin^{2}{\left(2 x \right)} = 0$$
transform
$$1 - 2 \sin^{2}{\left(2 x \right)} = 0$$
$$1 - 2 \sin^{2}{\left(2 x \right)} = 0$$
Do replacement
$$w = \sin{\left(2 x \right)}$$
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
$$a = -2$$
$$b = 0$$
$$c = 1$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = - \frac{\sqrt{2}}{2}$$
$$w_{2} = \frac{\sqrt{2}}{2}$$
do backward replacement
$$\sin{\left(2 x \right)} = w$$
Given the equation
$$\sin{\left(2 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$2 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
substitute w:
$$x_{1} = \pi n + \frac{\operatorname{asin}{\left(w_{1} \right)}}{2}$$
$$x_{1} = \pi n + \frac{\operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}}{2}$$
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{\operatorname{asin}{\left(w_{2} \right)}}{2}$$
$$x_{2} = \pi n + \frac{\operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}}{2}$$
$$x_{2} = \pi n + \frac{\pi}{8}$$
$$x_{3} = \pi n - \frac{\operatorname{asin}{\left(w_{1} \right)}}{2} + \frac{\pi}{2}$$
$$x_{3} = \pi n - \frac{\operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}}{2} + \frac{\pi}{2}$$
$$x_{3} = \pi n + \frac{5 \pi}{8}$$
$$x_{4} = \pi n - \frac{\operatorname{asin}{\left(w_{2} \right)}}{2} + \frac{\pi}{2}$$
$$x_{4} = \pi n - \frac{\operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}}{2} + \frac{\pi}{2}$$
$$x_{4} = \pi n + \frac{3 \pi}{8}$$
$$1 - 2 \sin^{2}{\left(2 x \right)} = 0$$
transform
$$1 - 2 \sin^{2}{\left(2 x \right)} = 0$$
$$1 - 2 \sin^{2}{\left(2 x \right)} = 0$$
Do replacement
$$w = \sin{\left(2 x \right)}$$
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
$$a = -2$$
$$b = 0$$
$$c = 1$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (-2) * (1) = 8
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = - \frac{\sqrt{2}}{2}$$
$$w_{2} = \frac{\sqrt{2}}{2}$$
do backward replacement
$$\sin{\left(2 x \right)} = w$$
Given the equation
$$\sin{\left(2 x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$2 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$2 x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$2 x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
substitute w:
$$x_{1} = \pi n + \frac{\operatorname{asin}{\left(w_{1} \right)}}{2}$$
$$x_{1} = \pi n + \frac{\operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}}{2}$$
$$x_{1} = \pi n - \frac{\pi}{8}$$
$$x_{2} = \pi n + \frac{\operatorname{asin}{\left(w_{2} \right)}}{2}$$
$$x_{2} = \pi n + \frac{\operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}}{2}$$
$$x_{2} = \pi n + \frac{\pi}{8}$$
$$x_{3} = \pi n - \frac{\operatorname{asin}{\left(w_{1} \right)}}{2} + \frac{\pi}{2}$$
$$x_{3} = \pi n - \frac{\operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}}{2} + \frac{\pi}{2}$$
$$x_{3} = \pi n + \frac{5 \pi}{8}$$
$$x_{4} = \pi n - \frac{\operatorname{asin}{\left(w_{2} \right)}}{2} + \frac{\pi}{2}$$
$$x_{4} = \pi n - \frac{\operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}}{2} + \frac{\pi}{2}$$
$$x_{4} = \pi n + \frac{3 \pi}{8}$$
Rapid solution
[src]
-pi
x1 = ----
8
$$x_{1} = - \frac{\pi}{8}$$
pi
x2 = --
8
$$x_{2} = \frac{\pi}{8}$$
3*pi
x3 = ----
8
$$x_{3} = \frac{3 \pi}{8}$$
5*pi
x4 = ----
8
$$x_{4} = \frac{5 \pi}{8}$$
x4 = 5*pi/8
Sum and product of roots
[src]
sum
pi pi 3*pi 5*pi - -- + -- + ---- + ---- 8 8 8 8
$$\left(\left(- \frac{\pi}{8} + \frac{\pi}{8}\right) + \frac{3 \pi}{8}\right) + \frac{5 \pi}{8}$$
=
pi
$$\pi$$
product
-pi pi 3*pi 5*pi ----*--*----*---- 8 8 8 8
$$\frac{5 \pi}{8} \frac{3 \pi}{8} \cdot - \frac{\pi}{8} \frac{\pi}{8}$$
=
4 -15*pi ------- 4096
$$- \frac{15 \pi^{4}}{4096}$$
-15*pi^4/4096
Numerical answer
[src]
x1 = 67.9369411338793
x2 = -2.74889357189107
x3 = 52.2289778659303
x4 = -13.7444678594553
x5 = 82.0741080750334
x6 = 44.3749962319558
x7 = 1.96349540849362
x8 = 27.8816348006094
x9 = -64.009950316892
x10 = -20.0276531666349
x11 = -49.872783375738
x12 = 23.9546439836222
x13 = 22.3838476568273
x14 = 60.0829594999048
x15 = 115.846229101124
x16 = 0.392699081698724
x17 = -9.8174770424681
x18 = 96.2112750161874
x19 = -56.1559686829176
x20 = 4.31968989868597
x21 = 42.0188017417635
x22 = 56.1559686829176
x23 = -67.9369411338793
x24 = 93.8550805259951
x25 = 54.5851723561227
x26 = -100.138265833175
x27 = 74.2201264410589
x28 = -12.1736715326604
x29 = -89.9280897090078
x30 = 12.1736715326604
x31 = -79.717913584841
x32 = 104.850654813559
x33 = -65.5807466436869
x34 = 92.2842841992002
x35 = -87.5718952188155
x36 = 70.2931356240716
x37 = 45.9457925587507
x38 = -34.164820107789
x39 = -5.89048622548086
x40 = 11492.3386259132
x41 = -43.5895980685584
x42 = 48.3019870489431
x43 = -1.96349540849362
x44 = 71.8639319508665
x45 = -57.7267650097125
x46 = 49.872783375738
x47 = -17.6714586764426
x48 = -35.7356164345839
x49 = -86.0010988920206
x50 = 64.009950316892
x51 = 53.7997741927252
x52 = -78.1471172580461
x53 = -82.0741080750334
x54 = 30.2378292908018
x55 = 89.9280897090078
x56 = -21.5984494934298
x57 = -75.0055246044563
x58 = -16.1006623496477
x59 = 62.4391539900971
x60 = -27.8816348006094
x61 = -60.0829594999048
x62 = -93.8550805259951
x63 = 8.24668071567321
x64 = 84.4303025652257
x65 = 9.8174770424681
x66 = 88.3572933822129
x67 = -97.7820713429823
x68 = -20.8130513300324
x69 = 38.0918109247762
x70 = -23.9546439836222
x71 = 26.3108384738145
x72 = -75.7909227678538
x73 = -10.6028752058656
x74 = 40.4480054149686
x75 = -31.8086256175967
x76 = -53.7997741927252
x77 = 66.3661448070844
x78 = -38.0918109247762
x79 = 16.1006623496477
x80 = -39.6626072515711
x81 = 96.9966731795849
x82 = 34.164820107789
x83 = -96.9966731795849
x84 = -83.6449044018282
x85 = -68.7223392972767
x86 = -42.0188017417635
x87 = 86.0010988920206
x88 = -45.9457925587507
x89 = 20.0276531666349
x90 = -71.8639319508665
x91 = 5.89048622548086
x92 = 31.8086256175967
x93 = -61.6537558266997
x94 = 100.138265833175
x95 = 78.1471172580461
x95 = 78.1471172580461