Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation 2tg3x=0
-
Equation cos(x+pi/2)=sqrt(3)/2
- Equation 8cos^2(x)+6sin(x)-3=0
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Equation sin((pi(8x+3)/6))=0,5
- 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
- Derivative of:
-
0,5
- Sum of series:
-
0,5
- Integral of d{x}:
- 0,5
-
Identical expressions
- sin((pi(8x+ three)/ six))= zero , five
- sinus of (( Pi (8x plus 3) divide by 6)) equally 0,5
- sinus of (( Pi (8x plus three) divide by six)) equally zero , five
- sinpi8x+3/6=0,5
- sin((pi(8x+3)/6))=O,5
- sin((pi(8x+3) divide by 6))=0,5
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Similar expressions
- sin((pi(8x-3)/6))=0,5
- sin((pi*(8*x+3)/6))=-sqrt(3)/2
sin((pi(8x+3)/6))=0,5 equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
/pi*(8*x + 3)\ sin|------------| = 1/2 \ 6 /
$$\sin{\left(\frac{\pi \left(8 x + 3\right)}{6} \right)} = \frac{1}{2}$$
Detail solution
Given the equation
$$\sin{\left(\frac{\pi \left(8 x + 3\right)}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{4 \pi x}{3} = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{4 \pi x}{3} = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{4 \pi x}{3} = 2 \pi n + \frac{\pi}{3}$$
$$\frac{4 \pi x}{3} = 2 \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{4 \pi}{3}$$
we get the answer:
$$x_{1} = \frac{3 \cdot \left(2 \pi n + \frac{\pi}{3}\right)}{4 \pi}$$
$$x_{2} = \frac{3 \cdot \left(2 \pi n - \frac{2 \pi}{3}\right)}{4 \pi}$$
$$\sin{\left(\frac{\pi \left(8 x + 3\right)}{6} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{4 \pi x}{3} = 2 \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{4 \pi x}{3} = 2 \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{4 \pi x}{3} = 2 \pi n + \frac{\pi}{3}$$
$$\frac{4 \pi x}{3} = 2 \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{4 \pi}{3}$$
we get the answer:
$$x_{1} = \frac{3 \cdot \left(2 \pi n + \frac{\pi}{3}\right)}{4 \pi}$$
$$x_{2} = \frac{3 \cdot \left(2 \pi n - \frac{2 \pi}{3}\right)}{4 \pi}$$
Sum and product of roots
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sum
1/4 + 5/4
$$\left(\frac{1}{4}\right) + \left(\frac{5}{4}\right)$$
=
3/2
$$\frac{3}{2}$$
product
1/4 * 5/4
$$\left(\frac{1}{4}\right) * \left(\frac{5}{4}\right)$$
=
5/16
$$\frac{5}{16}$$
Numerical answer
[src]
x1 = 28.25
x2 = 42.25
x3 = 24.25
x4 = 85.75
x5 = 7.75
x6 = 0.25
x7 = 25.75
x8 = 82.25
x9 = -91.75
x10 = 37.75
x11 = 34.25
x12 = 54.25
x13 = -52.25
x14 = 18.25
x15 = 13.75
x16 = -29.75
x17 = -46.25
x18 = 78.25
x19 = -37.75
x20 = 67.75
x21 = 49.75
x22 = -35.75
x23 = 48.25
x24 = -23.75
x25 = 90.25
x26 = 64.25
x27 = -10.25
x28 = -67.75
x29 = -94.25
x30 = 10.25
x31 = -40.25
x32 = -58.25
x33 = 58.25
x34 = 72.25
x35 = -71.75
x36 = 60.25
x37 = -82.25
x38 = -28.25
x39 = 31.75
x40 = 1.75
x41 = -79.75
x42 = -97.75
x43 = -85.75
x44 = -22.25
x45 = 96.25
x46 = 12.25
x47 = -49.75
x48 = 46.25
x49 = -61.75
x50 = -65.75
x51 = -19.75
x52 = -64.25
x53 = -4.25
x54 = 84.25
x55 = -55.75
x56 = 43.75
x57 = -95.75
x58 = -34.25
x59 = -7.75
x60 = 61.75
x61 = -25.75
x62 = 52.25
x63 = -5.75
x64 = 19.75
x65 = -11.75
x66 = 55.75
x67 = -89.75
x68 = -1.75
x69 = -100.25
x70 = 88.25
x71 = -76.25
x72 = -43.75
x73 = 79.75
x74 = 16.25
x75 = -73.75
x76 = 6.25
x77 = 70.25
x78 = -83.75
x79 = -13.75
x80 = -16.25
x81 = 91.75
x82 = -17.75
x83 = 100.25
x84 = -41.75
x85 = -31.75
x86 = 76.25
x87 = 36.25
x88 = -59.75
x89 = -53.75
x90 = 66.25
x91 = 22.25
x92 = -47.75
x93 = 97.75
x94 = -70.25
x95 = 40.25
x96 = -77.75
x97 = 73.75
x98 = 94.25
x99 = -88.25
x100 = 4.25
x101 = 30.25
x101 = 30.25
The graph