Mister Exam
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How to use it?
- Equation:
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Equation 2x^2-3x+1=0
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Equation x^2-4x+13=0
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Equation x^3+x^2-9*x-9=0
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Equation 7/(x-5)=2
- Express {x} in terms of y where:
- 5*x-4*y=14
- -7*x+15*y=7
- -11*x+3*y=5
- -13*x+13*y=-20
-
Identical expressions
- cos(three *x-pi/ four)+sqrt(two)/ two = zero
- co sinus of e of (3 multiply by x minus Pi divide by 4) plus square root of (2) divide by 2 equally 0
- co sinus of e of (three multiply by x minus Pi divide by four) plus square root of (two) divide by two equally zero
- cos(3*x-pi/4)+√(2)/2=0
- cos(3x-pi/4)+sqrt(2)/2=0
- cos3x-pi/4+sqrt2/2=0
- cos(3*x-pi/4)+sqrt(2)/2=O
- cos(3*x-pi divide by 4)+sqrt(2) divide by 2=0
-
Similar expressions
- cos(3*x-pi/4)-sqrt(2)/2=0
- cos(3*x+pi/4)+sqrt(2)/2=0
cos(3*x-pi/4)+sqrt(2)/2=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
___ / pi\ \/ 2 cos|3*x - --| + ----- = 0 \ 4 / 2
$$\cos{\left(3 x - \frac{\pi}{4} \right)} + \frac{\sqrt{2}}{2} = 0$$
Detail solution
Given the equation
$$\cos{\left(3 x - \frac{\pi}{4} \right)} + \frac{\sqrt{2}}{2} = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(3 x - \frac{\pi}{4} \right)} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = - \frac{\sqrt{2}}{2}$$
This equation is transformed to
$$3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$3 x + \frac{\pi}{4} = 2 \pi n - \frac{\pi}{4}$$
$$3 x + \frac{\pi}{4} = 2 \pi n + \frac{5 \pi}{4}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n - \frac{\pi}{2}$$
$$3 x = 2 \pi n + \pi$$
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{6}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
$$\cos{\left(3 x - \frac{\pi}{4} \right)} + \frac{\sqrt{2}}{2} = 0$$
- this is the simplest trigonometric equation
Move sqrt(2)/2 to right part of the equation
with the change of sign in sqrt(2)/2
We get:
$$\cos{\left(3 x - \frac{\pi}{4} \right)} - \frac{\sqrt{2}}{2} + \frac{\sqrt{2}}{2} = - \frac{\sqrt{2}}{2}$$
This equation is transformed to
$$3 x + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)}$$
$$3 x + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(- \frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$3 x + \frac{\pi}{4} = 2 \pi n - \frac{\pi}{4}$$
$$3 x + \frac{\pi}{4} = 2 \pi n + \frac{5 \pi}{4}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$3 x = 2 \pi n - \frac{\pi}{2}$$
$$3 x = 2 \pi n + \pi$$
Divide both parts of the equation by
$$3$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{3} - \frac{\pi}{6}$$
$$x_{2} = \frac{2 \pi n}{3} + \frac{\pi}{3}$$
Rapid solution
[src]
-pi
x1 = ----
6
$$x_{1} = - \frac{\pi}{6}$$
pi
x2 = --
3
$$x_{2} = \frac{\pi}{3}$$
x2 = pi/3
Sum and product of roots
[src]
sum
pi pi - -- + -- 6 3
$$- \frac{\pi}{6} + \frac{\pi}{3}$$
=
pi -- 6
$$\frac{\pi}{6}$$
product
-pi pi ----*-- 6 3
$$- \frac{\pi}{6} \frac{\pi}{3}$$
=
2 -pi ----- 18
$$- \frac{\pi^{2}}{18}$$
-pi^2/18
Numerical answer
[src]
x1 = -55.5014702134197
x2 = 62.3082542961976
x3 = -86.3937979737193
x4 = -1.0471975511966
x5 = 14.1371669411541
x6 = -80.1106126665397
x7 = -69.6386371545737
x8 = -4.71238898038469
x9 = -34.0339204138894
x10 = 74.3510261349584
x11 = -59.6902604182061
x12 = -31.9395253114962
x13 = 93.7241808320955
x14 = 26.1799387799149
x15 = 18.3259571459405
x16 = 95.8185759344887
x17 = 72.2566310325652
x18 = 70.162235930172
x19 = -27.7507351067098
x20 = -24.0855436775217
x21 = 7.85398163397448
x22 = -95.2949771588904
x23 = -29.845130209103
x24 = 80.634211442138
x25 = 49.7418836818384
x26 = -6.80678408277789
x27 = 39.2699081698724
x28 = 47.6474885794452
x29 = 30.3687289847013
x30 = 100.007366139275
x31 = -53.4070751110265
x32 = 45.553093477052
x33 = -21.9911485751286
x34 = -71.733032256967
x35 = 58.1194640914112
x36 = 32.4631240870945
x37 = -78.0162175641465
x38 = -36.1283155162826
x39 = -228.812664936457
x40 = -73.8274273593601
x41 = -19.8967534727354
x42 = -11.5191730631626
x43 = -42.4115008234622
x44 = 53.9306738866248
x45 = 16.2315620435473
x46 = -65.9734457253857
x47 = 83.2522053201295
x48 = 1.5707963267949
x49 = 51.3126800086333
x50 = -75.9218224617533
x51 = -82.2050077689329
x52 = 60.2138591938044
x53 = -99.4837673636768
x54 = -82.7286065445312
x55 = 76.4454212373516
x56 = -17.8023583703422
x57 = 28.2743338823081
x58 = -68.0678408277789
x59 = -13.6135681655558
x60 = -38.2227106186758
x61 = 97.9129710368819
x62 = -7.33038285837618
x63 = -45.0294947014537
x64 = -9.42477796076938
x65 = 65.9734457253857
x66 = 12.0427718387609
x67 = -80.634211442138
x68 = 154.461638801498
x69 = -25.6563400043166
x70 = -15.707963267949
x71 = -89.0117918517108
x72 = 36.6519142918809
x73 = -48.6946861306418
x74 = 5.75958653158129
x75 = -51.3126800086333
x76 = -84.2994028713261
x77 = 3.66519142918809
x78 = 51.8362787842316
x79 = -57.5958653158129
x80 = 24.0855436775217
x81 = 9.94837673636768
x82 = -61.7846555205993
x83 = 21.9911485751286
x84 = 34.5575191894877
x85 = -97.3893722612836
x86 = 91.6297857297023
x87 = 78.5398163397448
x88 = -40.317105721069
x89 = 56.025068989018
x90 = -63.8790506229925
x91 = 68.0678408277789
x92 = 19.8967534727354
x92 = 19.8967534727354