Mister Exam
Other calculators
-
How to use it?
- Equation:
- Equation 1-y^2=2+x
-
Equation 1-7*(4+2*x)=-9-4*x
-
Equation 2*cos(x)+1=0
-
Equation 3x-2=5
- Express {x} in terms of y where:
- 17*x-18*y=10
- -7*x+7*y=-14
- 10*x-16*y=6
- 1*x-4*y=1
- Integral of d{x}:
-
sin(5x)
- Derivative of:
-
sin(5x)
-
Identical expressions
- sin(5x)=√ two \ two
- sinus of (5x) equally √2\2
- sinus of (5x) equally √ two \ two
- sin5x=√2\2
-
Similar expressions
- (2cosx/2-sqrt2)*(sin5x+2)=0
- 2sin5x=1
sin(5x)=√2\2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(5 x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$5 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$5 x = 2 \pi n + \frac{\pi}{4}$$
$$5 x = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$5$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{5} + \frac{\pi}{20}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{3 \pi}{20}$$
$$\sin{\left(5 x \right)} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$5 x = 2 \pi n + \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)}$$
$$5 x = 2 \pi n - \operatorname{asin}{\left(\frac{\sqrt{2}}{2} \right)} + \pi$$
Or
$$5 x = 2 \pi n + \frac{\pi}{4}$$
$$5 x = 2 \pi n + \frac{3 \pi}{4}$$
, where n - is a integer
Divide both parts of the equation by
$$5$$
we get the answer:
$$x_{1} = \frac{2 \pi n}{5} + \frac{\pi}{20}$$
$$x_{2} = \frac{2 \pi n}{5} + \frac{3 \pi}{20}$$
Rapid solution
[src]
pi
x1 = --
20
$$x_{1} = \frac{\pi}{20}$$
3*pi
x2 = ----
20
$$x_{2} = \frac{3 \pi}{20}$$
Sum and product of roots
[src]
sum
pi 3*pi
0 + -- + ----
20 20
$$\left(0 + \frac{\pi}{20}\right) + \frac{3 \pi}{20}$$
=
pi -- 5
$$\frac{\pi}{5}$$
product
pi 3*pi 1*--*---- 20 20
$$\frac{3 \pi}{20} \cdot 1 \frac{\pi}{20}$$
=
2 3*pi ----- 400
$$\frac{3 \pi^{2}}{400}$$
3*pi^2/400
Numerical answer
[src]
x1 = -30.0022098417825
x2 = -91.5774258521425
x3 = 24.0331837999619
x4 = -53.564154743706
x5 = -23.7190245346029
x6 = 44.1393767829366
x7 = -66.1305253580651
x8 = -97.8606111593221
x9 = -69.9004365423729
x10 = 99.431407486117
x11 = -49.7942435593982
x12 = -67.70132168486
x13 = -2.04203522483337
x14 = 51.9933584169111
x15 = 84.3517627488859
x16 = 20.2632726156542
x17 = -12.0951317163207
x18 = 40.3694655986288
x19 = 68.015480950219
x20 = -68.957958746296
x21 = 6.44026493985908
x22 = -33.7721210260903
x23 = -9.89601685880785
x24 = 48.2234472326033
x25 = -42.5685804561417
x26 = 64.2455697659113
x27 = -32.2013246992954
x28 = -39.7411470679109
x29 = 88.1216739331937
x30 = -87.8075146678347
x31 = 28.1172542496287
x32 = -5.81194640914112
x33 = -77.7544181763474
x34 = -87.4933554024757
x35 = -73.9845069920396
x36 = -90.0066295253476
x37 = 62.0464549083984
x38 = 58.2765437240907
x39 = 98.174770424681
x40 = 95.9756555671682
x41 = -81.5243293606551
x42 = -37.542032210398
x43 = 0.15707963267949
x44 = 60.4756585816035
x45 = -56.0774288665778
x46 = -13.6659280431156
x47 = -46.0243323750905
x48 = 72.0995513998858
x49 = -3.61283155162826
x50 = -83.723444218168
x51 = 74.2986662573986
x52 = -43.8252175175776
x53 = 75.8694625841935
x54 = 50.4225620901162
x55 = -100.059726016835
x56 = 65.8163660927062
x57 = 11.7809724509617
x58 = 21.8340689424491
x59 = 31.8871654339364
x60 = -47.5951287018854
x61 = 10.2101761241668
x62 = -22.148228207808
x63 = -59.8473400508856
x64 = 15.2367243699105
x65 = 16.8075206967054
x66 = 88.4358331985527
x67 = -79.9535330338602
x68 = 8053.0015285794
x69 = 4.24115008234622
x70 = -93.7765407096553
x71 = -76.1836218495525
x72 = 85.9225590756809
x73 = 78.0685774417064
x74 = 8.01106126665397
x75 = 1107.56849002308
x76 = -19.9491133502952
x77 = -57.6482251933727
x78 = 18.0641577581413
x79 = 41.9402619254237
x80 = -27.1747764535517
x81 = -53.878314009065
x82 = 54.5066325397829
x83 = 92.2057443828604
x84 = 82.1526478913731
x85 = 30.3163691071415
x86 = 38.170350741116
x87 = -10.8384946548848
x88 = 94.4048592403733
x89 = -35.9712358836031
x90 = 54.1924732744239
x91 = 34.0862802914493
x92 = -25.9181393921158
x93 = -15.8650429006285
x94 = -63.9314105005523
x94 = -63.9314105005523
The graph