Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (x-5)/2=2*(3x/7-1/2)/3
-
Equation 2*x^2+6*x+9=0
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Equation cos(x/2)=(1/2)
-
Equation (x+2)^2=(1-x)^2
- Express {x} in terms of y where:
- 17*x-18*y=10
- 9*x-4*y=2
- 20*x+6*y=8
- -15*x+17*y=13
- Derivative of:
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cos(x/2)
- (1/2)
- Graphing y =:
- cos(x/2)
- Limit of the function:
-
cos(x/2)
- Sum of series:
-
(1/2)
-
Identical expressions
- cos(x/ two)=(one / two)
- co sinus of e of (x divide by 2) equally (1 divide by 2)
- co sinus of e of (x divide by two) equally (one divide by two)
- cosx/2=1/2
- cos(x divide by 2)=(1 divide by 2)
-
Similar expressions
- sin(5x)=-1/2
- (3)/(8y^3+1)-(1)/(2y+1)=(y+3)/(y^2-2y+1)
- (x-5)/2=2*(3x/7-1/2)/3
- cos(x/2-pi/6)=1
- sinx+(cosx/2-sinx/2)(cox/2+sinx/2)=0
- cos(4pix/3)=-1/2
cos(x/2)=(1/2) equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\cos{\left(\frac{x}{2} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{x}{2} = \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{2 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{4 \pi}{3}$$
$$\cos{\left(\frac{x}{2} \right)} = \frac{1}{2}$$
- this is the simplest trigonometric equation
This equation is transformed to
$$\frac{x}{2} = \pi n + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
$$\frac{x}{2} = \pi n - \pi + \operatorname{acos}{\left(\frac{1}{2} \right)}$$
Or
$$\frac{x}{2} = \pi n + \frac{\pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{2 \pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{2 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{4 \pi}{3}$$
Rapid solution
[src]
2*pi
x1 = ----
3
$$x_{1} = \frac{2 \pi}{3}$$
10*pi
x2 = -----
3
$$x_{2} = \frac{10 \pi}{3}$$
Sum and product of roots
[src]
sum
2*pi 10*pi
0 + ---- + -----
3 3
$$\left(0 + \frac{2 \pi}{3}\right) + \frac{10 \pi}{3}$$
=
4*pi
$$4 \pi$$
product
2*pi 10*pi 1*----*----- 3 3
$$\frac{10 \pi}{3} \cdot 1 \cdot \frac{2 \pi}{3}$$
=
2 20*pi ------ 9
$$\frac{20 \pi^{2}}{9}$$
20*pi^2/9
Numerical answer
[src]
x1 = -85.870199198121
x2 = -14.6607657167524
x3 = -39.7935069454707
x4 = 2.0943951023932
x5 = 73.3038285837618
x6 = -111.002940426839
x7 = -48.1710873550435
x8 = -2.0943951023932
x9 = 85.870199198121
x10 = 23.0383461263252
x11 = 64.9262481741891
x12 = -90.0589894029074
x13 = -429.350995990605
x14 = 52.3598775598299
x15 = 39.7935069454707
x16 = 60.7374579694027
x17 = 48.1710873550435
x18 = 98.4365698124802
x19 = -23.0383461263252
x20 = -27.2271363311115
x21 = -77.4926187885482
x22 = -64.9262481741891
x23 = -60.7374579694027
x24 = 90.0589894029074
x25 = 77.4926187885482
x26 = -278.554548618295
x27 = -437.728576400178
x28 = -98.4365698124802
x29 = 10.471975511966
x30 = -73.3038285837618
x31 = -10.471975511966
x32 = -265014.284281323
x33 = -35.6047167406843
x34 = 35.6047167406843
x35 = 2088.11191708602
x36 = -605.280184591634
x37 = 27.2271363311115
x38 = 14.6607657167524
x39 = -52.3598775598299
x39 = -52.3598775598299
The graph