Mister Exam
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Identical expressions
- cos((x/ two)-(pi/ four))= one
- co sinus of e of ((x divide by 2) minus ( Pi divide by 4)) equally 1
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- cosx/2-pi/4=1
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-
Similar expressions
- cos((x/2)+(pi/4))=1
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- cos((x)/(2)-(pi)/(4))=-1
cos((x/2)-(pi/4))=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x pi\ cos|- - --| = 1 \2 4 /
$$\cos{\left(\frac{x}{2} - \frac{\pi}{4} \right)} = 1$$
Detail solution
Given the equation
$$\cos{\left(\frac{x}{2} - \frac{\pi}{4} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{4}$$
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{2}$$
$$x_{2} = 4 \pi n + \frac{\pi}{2}$$
$$\cos{\left(\frac{x}{2} - \frac{\pi}{4} \right)} = 1$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \operatorname{asin}{\left(1 \right)}$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n - \operatorname{asin}{\left(1 \right)} + \pi$$
Or
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{2}$$
$$\frac{x}{2} + \frac{\pi}{4} = 2 \pi n + \frac{\pi}{2}$$
, where n - is a integer
Move
$$\frac{\pi}{4}$$
to right part of the equation
with the opposite sign, in total:
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{4}$$
$$\frac{x}{2} = 2 \pi n + \frac{\pi}{4}$$
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{\pi}{2}$$
$$x_{2} = 4 \pi n + \frac{\pi}{2}$$
Sum and product of roots
[src]
sum
pi -- 2
$$\frac{\pi}{2}$$
=
pi -- 2
$$\frac{\pi}{2}$$
product
pi -- 2
$$\frac{\pi}{2}$$
=
pi -- 2
$$\frac{\pi}{2}$$
pi/2
Numerical answer
[src]
x1 = 102.101760444762
x2 = -36.1283148380364
x3 = 39.2699071461298
x4 = 26.7035366020848
x5 = 1.57079657962028
x6 = -98.9601689788754
x7 = -98.96016809972
x8 = 64.4026485148034
x9 = 51.836277811728
x10 = -48.6946836170949
x11 = -36.128316479355
x12 = 14.1371677603771
x13 = -10.9955732565122
x14 = 14.1371671112396
x15 = -10.9955737844476
x16 = 39.269907680759
x17 = -48.6946866505019
x18 = -36.1283146222806
x19 = 64.4026487448103
x20 = -10.99557350544
x21 = 26.7035385797517
x22 = 76.9690190152848
x23 = 1.57079571674813
x24 = 76.9690205307018
x25 = 26.7035373545977
x26 = -23.5619439203143
x27 = -10.9955752897303
x28 = 76.9690206347008
x29 = 14.1371660851599
x30 = -73.8274264713758
x31 = -23.5619441812655
x32 = 64.4026503711459
x33 = 76.9690175844797
x34 = 89.5353915964849
x35 = -86.3937970183568
x36 = -36.1283162369255
x37 = -36.1283154168665
x38 = 14.1371678113613
x39 = 51.8362796872383
x40 = 102.101760367026
x41 = -23.5619450121181
x42 = 64.4026501315336
x43 = 39.2699091799454
x44 = 89.5353896076556
x45 = 39.2699085594554
x46 = 1.57079657820027
x47 = -73.8274272797971
x48 = -48.6946857566901
x49 = 89.5353900309945
x50 = -73.8274265510174
x51 = 76.9690196533244
x52 = 89.5353908041884
x53 = -61.2610568665699
x54 = 1.57079531004407
x55 = -73.8274281950731
x56 = -23.5619455179347
x57 = -86.3937976542222
x58 = -98.9601675585148
x59 = -48.6946871554707
x60 = -98.9601705926202
x61 = -61.261057705332
x62 = 1.57079728920695
x63 = -61.2610561353103
x64 = -48.6946851249588
x65 = -36.1283139903096
x66 = 64.4026493068485
x67 = -61.2610570004552
x68 = 51.836279426748
x69 = -10.9955746628274
x70 = -86.3937977376697
x71 = 26.703537316508
x72 = 19718.2062873233
x73 = 51.8362789039323
x74 = -10.9955765216595
x75 = 14.1371670569947
x76 = -48.6946866344595
x77 = -86.3937989869347
x78 = -73.8274281781036
x79 = 14.1371661332277
x80 = 39.2699074996103
x81 = 26.7035381788358
x82 = -23.5619457948623
x83 = 51.8362817117123
x84 = -61.2610557179513
x85 = -23.5619447685186
x86 = 76.9690210450504
x87 = -86.3937985973555
x88 = 51.8362780759985
x89 = 89.5353908967368
x90 = 39.2699105457115
x90 = 39.2699105457115