Mister Exam
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How to use it?
- Equation:
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Equation 1-7*(4+2*x)=-9-4*x
-
Equation sin(x)=2
-
Equation -x-2+3(x-3)=3(4-x)-3
-
Equation 4x=24+x
- Express {x} in terms of y where:
- -6*x-2*y=-10
- 18*x+14*y=-11
- -8*x-12*y=15
- sin(xy)+cos(xy)=tg(x+y)
- Derivative of:
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sin(x/2)
- Graphing y =:
- sin(x/2)
- Integral of d{x}:
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sin(x/2)
-
Identical expressions
- sin(x/ two)=cos(pi/ eight)
- sinus of (x divide by 2) equally co sinus of e of ( Pi divide by 8)
- sinus of (x divide by two) equally co sinus of e of ( Pi divide by eight)
- sinx/2=cospi/8
- sin(x divide by 2)=cos(pi divide by 8)
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Similar expressions
- sinx*cospi/8+cosx*sinpi/8=1/2
- 2*sin(x)/2=1-cos(x)
- (2sin^2*x-√3sinx)/(2cosx+1)=0
sin(x/2)=cos(pi/8) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x\ /pi\ sin|-| = cos|--| \2/ \8 /
$$\sin{\left(\frac{x}{2} \right)} = \cos{\left(\frac{\pi}{8} \right)}$$
Detail solution
Given the equation
$$\sin{\left(\frac{x}{2} \right)} = \cos{\left(\frac{\pi}{8} \right)}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n + \frac{3 \pi}{8}$$
$$\frac{x}{2} = 2 \pi n + \frac{5 \pi}{8}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{3 \pi}{4}$$
$$x_{2} = 4 \pi n + \frac{5 \pi}{4}$$
$$\sin{\left(\frac{x}{2} \right)} = \cos{\left(\frac{\pi}{8} \right)}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = 2 \pi n + \operatorname{asin}{\left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \right)}$$
$$\frac{x}{2} = 2 \pi n - \operatorname{asin}{\left(\sqrt{\frac{\sqrt{2}}{4} + \frac{1}{2}} \right)} + \pi$$
Or
$$\frac{x}{2} = 2 \pi n + \frac{3 \pi}{8}$$
$$\frac{x}{2} = 2 \pi n + \frac{5 \pi}{8}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 4 \pi n + \frac{3 \pi}{4}$$
$$x_{2} = 4 \pi n + \frac{5 \pi}{4}$$
Rapid solution
[src]
3*pi
x1 = ----
4
$$x_{1} = \frac{3 \pi}{4}$$
5*pi
x2 = ----
4
$$x_{2} = \frac{5 \pi}{4}$$
Sum and product of roots
[src]
sum
3*pi 5*pi
0 + ---- + ----
4 4
$$\left(0 + \frac{3 \pi}{4}\right) + \frac{5 \pi}{4}$$
=
2*pi
$$2 \pi$$
product
3*pi 5*pi 1*----*---- 4 4
$$\frac{5 \pi}{4} \cdot 1 \cdot \frac{3 \pi}{4}$$
=
2 15*pi ------ 16
$$\frac{15 \pi^{2}}{16}$$
15*pi^2/16
Numerical answer
[src]
x1 = -46.3384916404494
x2 = 52.621676947629
x3 = -96.6039740978861
x4 = 16.4933614313464
x5 = -47.9092879672443
x6 = 90.3207887907066
x7 = 102.887159405066
x8 = -110.74114103904
x9 = -12124.1914483664
x10 = 781.471172580461
x11 = -84.037603483527
x12 = -73.0420291959627
x13 = -58.9048622548086
x14 = 3.92699081698724
x15 = 2.35619449019234
x16 = -85.6083998103219
x17 = 253.683606777376
x18 = 77.7544181763474
x19 = -98.174770424681
x20 = -8.63937979737193
x21 = 40.0553063332699
x22 = -33.7721210260903
x23 = 41.6261026600648
x24 = 230.121661875452
x25 = -35.3429173528852
x26 = 27.4889357189107
x27 = -21.2057504117311
x28 = 91.8915851175014
x29 = 66.7588438887831
x30 = -22.776546738526
x31 = -10.2101761241668
x32 = 519.148186005713
x33 = 505.011019064559
x34 = 79.3252145031423
x35 = -60.4756585816035
x36 = 65.1880475619882
x37 = 29.0597320457056
x38 = 54.1924732744239
x39 = -71.4712328691678
x40 = 14.9225651045515
x40 = 14.9225651045515
The graph