Mister Exam
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cos(2*x)-1=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(2 x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
We get:
$$\cos{\left(2 x \right)} - 1 + 1 = 1$$
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$2 x = \pi n$$
$$2 x = \pi n - \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \frac{\pi n}{2}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{2}$$
$$\cos{\left(2 x \right)} - 1 = 0$$
- this is the simplest trigonometric equation
Move -1 to right part of the equation
with the change of sign in -1
We get:
$$\cos{\left(2 x \right)} - 1 + 1 = 1$$
This equation is transformed to
$$2 x = \pi n + \operatorname{acos}{\left(1 \right)}$$
$$2 x = \pi n - \pi + \operatorname{acos}{\left(1 \right)}$$
Or
$$2 x = \pi n$$
$$2 x = \pi n - \pi$$
, where n - is a integer
Divide both parts of the equation by
$$2$$
we get the answer:
$$x_{1} = \frac{\pi n}{2}$$
$$x_{2} = \frac{\pi n}{2} - \frac{\pi}{2}$$
Sum and product of roots
[src]
sum
0 + 0 + pi
$$\left(0 + 0\right) + \pi$$
=
pi
$$\pi$$
product
1*0*pi
$$1 \cdot 0 \pi$$
=
0
$$0$$
0
Numerical answer
[src]
x1 = 87.9645943357576
x2 = 53.4070753627408
x3 = -47.123890151099
x4 = -21.9911485864515
x5 = 75.3982241944528
x6 = -6371388.6625218
x7 = 28.2743338652012
x8 = 62.8318524523063
x9 = -15.7079632965264
x10 = 25.1327410188866
x11 = -25.132741632083
x12 = -53.4070752836338
x13 = 6.28318528425126
x14 = -91.106187201329
x15 = -43.9822971745789
x16 = -18.8495561207399
x17 = -65.9734457650176
x18 = -18.8495556944209
x19 = -106.814150357553
x20 = 47.123890018392
x21 = -47.1238900492539
x22 = -31.4159267051849
x23 = 9.42477859080277
x24 = 40.8407042560881
x25 = 34.5575190304759
x26 = -87.9645943587732
x27 = -34.5575189701076
x28 = -3.14159311568248
x29 = 53.4070756765307
x30 = -72.2566308741333
x31 = 18.8495556796107
x32 = -69.1150386737158
x33 = 84.8230014093114
x34 = 31.4159271479423
x35 = 62.8318528326557
x36 = -9.42477812668337
x37 = -12.5663700417108
x38 = 97.3893727097471
x39 = 84.8230010166547
x40 = -31.4159267959754
x41 = 25.1327414478072
x42 = 3.14159244884412
x43 = -50.2654822953391
x44 = 94.2477796093525
x45 = -78.5398160958028
x46 = 47.123889589354
x47 = -84.82300141007
x48 = -62.8318532583801
x49 = 56.5486676091327
x50 = 78.5398161878405
x51 = 50.2654824463473
x52 = -12.5663703661411
x53 = 72.256631027719
x54 = -25.132741473063
x55 = 15.7079634406648
x56 = 31.4159267865366
x57 = -34.5575189426108
x58 = 9.42477821024198
x59 = 69.1150381602162
x60 = 91.1061867314459
x61 = 40.840703919946
x62 = 97.3893725148693
x63 = -97.3893724403711
x64 = -62.8318528379059
x65 = 69.1150385885879
x66 = 43.982297169427
x67 = -100.530964672522
x68 = -59.6902604576401
x69 = 37.6991120192083
x70 = 65.9734457528975
x71 = 18.8495554002244
x72 = -1734.15914475848
x73 = 100.530964766599
x74 = 0.0
x75 = 59.6902605976901
x76 = -84.8230018263493
x77 = 12.5663704518704
x78 = -56.5486675191652
x79 = -37.6991118771514
x80 = -6.28318513794069
x81 = 3.14159287686128
x82 = -75.3982238620294
x83 = -40.8407042660168
x84 = 81.6814091761104
x85 = 21.9911485851964
x86 = -40.8407046898283
x87 = -94.2477794529919
x88 = 75.3982239388525
x89 = -69.1150386253436
x90 = -3.14159289677385
x91 = -81.6814090380061
x92 = -91.1061872003049
x93 = 91.1061871583643
x94 = -28.2743337166085
x94 = -28.2743337166085
The graph