Mister Exam
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How to use it?
- Equation:
- Equation (arcctg(x)-arctg(x))x^2-2x+arcctg(x)-arctg(x)
- Equation sinx+|cosx|=sin4x+cos2x
- Equation z^4-2z^2+4=0
-
Equation sin((pi(5x+15))/6)=1/2
- Express {x} in terms of y where:
- (x^2+y^2-1)^3+x^2*y^3=0
- -1*x-19*y=-10
- 5*x+5*y=20
- 4*x-16*y=6
- Sum of series:
-
1/2
- Integral of d{x}:
-
1/2
- Limit of the function:
-
1/2
-
Identical expressions
- sin((pi(5x+ fifteen))/ six)= one / two
- sinus of (( Pi (5x plus 15)) divide by 6) equally 1 divide by 2
- sinus of (( Pi (5x plus fifteen)) divide by six) equally one divide by two
- sinpi5x+15/6=1/2
- sin((pi(5x+15)) divide by 6)=1 divide by 2
-
Similar expressions
- sin*(pi(*5x+15)/6)=1/2
- sinpi(5*x+15)/6=1/2
- sin((pi(5x-15))/6)=1/2
sin((pi(5x+15))/6)=1/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
/pi*(5*x + 15)\ sin|-------------| = 1/2 \ 6 /
$$\sin{\left(\frac{\pi \left(5 x + 15\right)}{6} \right)} = \frac{1}{2}$$
Detail solution
Given the equation
$$\sin{\left(\frac{\pi \left(5 x + 15\right)}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{5 \pi x}{6} = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{5 \pi x}{6} = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{5 \pi x}{6} = 2 \pi n + \frac{\pi}{3}$$
$$\frac{5 \pi x}{6} = 2 \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{5 \pi}{6}$$
we get the answer:
$$x_{1} = \frac{6 \cdot \left(2 \pi n + \frac{\pi}{3}\right)}{5 \pi}$$
$$x_{2} = \frac{6 \cdot \left(2 \pi n - \frac{2 \pi}{3}\right)}{5 \pi}$$
$$\sin{\left(\frac{\pi \left(5 x + 15\right)}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{5 \pi x}{6} = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{5 \pi x}{6} = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{5 \pi x}{6} = 2 \pi n + \frac{\pi}{3}$$
$$\frac{5 \pi x}{6} = 2 \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{5 \pi}{6}$$
we get the answer:
$$x_{1} = \frac{6 \cdot \left(2 \pi n + \frac{\pi}{3}\right)}{5 \pi}$$
$$x_{2} = \frac{6 \cdot \left(2 \pi n - \frac{2 \pi}{3}\right)}{5 \pi}$$
Sum and product of roots
[src]
sum
2/5 + 2
$$\left(\frac{2}{5}\right) + \left(2\right)$$
=
12/5
$$\frac{12}{5}$$
product
2/5 * 2
$$\left(\frac{2}{5}\right) * \left(2\right)$$
=
4/5
$$\frac{4}{5}$$
Numerical answer
[src]
x1 = 38.0
x2 = -10.0
x3 = 26.0
x4 = -100.4
x5 = 7.6
x6 = 98.0
x7 = 206.8
x8 = 43.6
x9 = 110.8
x10 = 72.4
x11 = 48.4
x12 = -55.6
x13 = 34.0
x14 = 67.6
x15 = -14.0
x16 = -35.6
x17 = 0.4
x18 = -19.6
x19 = 50.0
x20 = 76.4
x21 = -86.0
x22 = 50.8
x23 = 96.4
x24 = 58.0
x25 = 70.0
x26 = -59.6
x27 = -16.4
x28 = 36.4
x29 = 60.4
x30 = -47.6
x31 = -62.0
x32 = -26.0
x33 = 74.8
x34 = 55.6
x35 = 24.4
x36 = -58.0
x37 = -76.4
x38 = -70.0
x39 = 79.6
x40 = 84.4
x41 = -88.4
x42 = 62.8
x43 = -43.6
x44 = -64.4
x45 = -83.6
x46 = -4.4
x47 = -95.6
x48 = 86.0
x49 = 218.8
x50 = -31.6
x51 = 74.0
x52 = 16.4
x53 = 12.4
x54 = 94.0
x55 = 100.4
x56 = -22.0
x57 = 86.8
x58 = 134.8
x59 = 38.8
x60 = 88.4
x61 = -2.0
x62 = 62.0
x63 = -82.0
x64 = 2.0
x65 = 40.4
x66 = 91.6
x67 = 52.4
x68 = -67.6
x69 = -28.4
x70 = -52.4
x71 = -11.6
x72 = -46.0
x73 = -40.4
x74 = 28.4
x75 = 182.8
x76 = -71.6
x77 = 122.8
x78 = -79.6
x79 = 10.0
x80 = 194.8
x81 = 146.8
x82 = 82.0
x83 = 98.8
x84 = -38.0
x85 = -94.0
x86 = -98.0
x87 = 158.8
x88 = -74.0
x89 = -34.0
x90 = 64.4
x91 = -23.6
x92 = -91.6
x93 = 19.6
x94 = 170.8
x95 = -50.0
x96 = 46.0
x97 = 31.6
x98 = 22.0
x99 = 14.0
x100 = 4.4
x101 = -7.6
x101 = -7.6
The graph