Mister Exam
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Equation 4x=24+x
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Identical expressions
- cos(x/ two -pi/ six)= one
- co sinus of e of (x divide by 2 minus Pi divide by 6) equally 1
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- cosx/2-pi/6=1
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-
Similar expressions
- cos(x/2+pi/6)=1
cos(x/2-pi/6)=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ cos|- - --| = 1 \2 6 /
$$\cos{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 1$$
Detail solution
Given the equation
$$\cos{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n + \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{6}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{3}$$
$$x_{2} = 4 \pi n + \frac{\pi}{3}$$
$$\cos{\left(\frac{x}{2} - \frac{\pi}{6} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{3} = 2 \pi n + \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{3}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{6}$$
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{6}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{3}$$
$$x_{2} = 4 \pi n + \frac{\pi}{3}$$
Sum and product of roots
[src]
sum
pi -- 3
$$\frac{\pi}{3}$$
=
pi -- 3
$$\frac{\pi}{3}$$
product
pi -- 3
$$\frac{\pi}{3}$$
=
pi -- 3
$$\frac{\pi}{3}$$
pi/3
Numerical answer
[src]
x1 = 38.7463104218346
x2 = -49.2182843132079
x3 = -86.9173971587402
x4 = 1.04719654407146
x5 = 63.8790496620044
x6 = 89.0117914732073
x7 = 13.6135683980913
x8 = 89.011792882489
x9 = 1.04719858039672
x10 = -11.5191720717926
x11 = 89.0117896636845
x12 = -99.483766365145
x13 = -74.3510251884844
x14 = 51.3126795183286
x15 = -11.5191739668321
x16 = 51.3126795017714
x17 = 63.8790498997893
x18 = -36.6519139369665
x19 = -24.0855446278783
x20 = 26.1799396714145
x21 = 63.879051515091
x22 = 126.710906642518
x23 = -49.2182838765694
x24 = 13.6135691194163
x25 = -49.2182851814066
x26 = -74.3510267748163
x27 = -262.846582448684
x28 = -86.9173984675651
x29 = 51.3126789836367
x30 = -36.6519153173964
x31 = 13.6135679727523
x32 = 76.4454211233792
x33 = -24.0855427947148
x34 = -36.6519132919576
x35 = -86.9173962789486
x36 = -49.2182849216693
x37 = 76.4454203425392
x38 = -61.7846546130394
x39 = 38.7463091354129
x40 = 51.312682533752
x41 = -74.3510259194413
x42 = 1.04719715718101
x43 = -24.0855436048037
x44 = 38.7463100011107
x45 = 101.578161853184
x46 = 51.3126810133596
x47 = 38.7463092901489
x48 = -49.218285874516
x49 = 26.1799379486384
x50 = 1.04719559598165
x51 = -11.5191731950537
x52 = 76.4454206296045
x53 = -74.3510257409001
x54 = 26.1799379589469
x55 = -36.6519148133015
x56 = 13.6135684984697
x57 = -11.5191723575735
x58 = 101.578161450002
x59 = 1.04719854751095
x60 = -74.3510271430162
x61 = 26.1799388484399
x62 = 76.4454219554321
x63 = 38.7463084325438
x64 = -24.0855444129256
x65 = 51.3126803793184
x66 = -86.9173955580436
x67 = 89.0117923517671
x68 = 89.0117926690924
x69 = -86.9173957221185
x70 = 13.6135675391105
x71 = 26.1799395859688
x72 = 63.8790513083079
x73 = -61.7846554920129
x74 = -61.7846563294489
x75 = 1.04719803646945
x76 = -11.5191736290977
x77 = -86.9173977601397
x78 = -99.4837666707462
x79 = -24.0855429598693
x80 = 63.8790507180149
x81 = -36.6519118750653
x82 = -99.4837675131561
x83 = -49.2182850008954
x84 = -61.7846547256668
x85 = -36.651914647967
x86 = 13.6135671531582
x87 = 76.4454222207315
x88 = -61.7846563521337
x89 = 89.011790848653
x89 = 89.011790848653