Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation e^x=0
-
Equation 1-5*x=-6*x+8
-
Equation 3-3*6+2=x
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Equation cos(x/2)=-(1/2)
- Express {x} in terms of y where:
- 10*x-2*y=0
- 2*x+1*y=-8
- 2*x+2*y=-6
- 12*x-13*y=-14
- Graphing y =:
- cos(x/2)
- Derivative of:
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cos(x/2)
- Limit of the function:
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cos(x/2)
-
Identical expressions
- cos(x/ two)=-(one / two)
- co sinus of e of (x divide by 2) equally minus (1 divide by 2)
- co sinus of e of (x divide by two) equally minus (one divide by two)
- cosx/2=-1/2
- cos(x divide by 2)=-(1 divide by 2)
-
Similar expressions
- 2cos*(x/2-pi/4)*cos(pi/3)=1
- -2^cos(x)*log(2)*sin(x)*atan(sqrt(x))+2^cos(x)/(2*sqrt(x)*(x+1))=0
- cos^2x-1/2=0
- cos(x/2)=+(1/2)
- log(y-1)/2-log(y+1)/2=Const-x
- sqrt(3)/2*cos(x)-1/2*sin(x)=cos3x
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
$$\frac{x}{2} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
$$\cos{\left(\frac{x}{2} \right)} = - \frac{1}{2}$$
- 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
$$\frac{x}{2} = \pi n + \frac{2 \pi}{3}$$
$$\frac{x}{2} = \pi n - \frac{\pi}{3}$$
, where n - is a integer
Divide both parts of the equation by
$$\frac{1}{2}$$
we get the answer:
$$x_{1} = 2 \pi n + \frac{4 \pi}{3}$$
$$x_{2} = 2 \pi n - \frac{2 \pi}{3}$$
Sum and product of roots
[src]
sum
4*pi 8*pi
0 + ---- + ----
3 3
$$\left(0 + \frac{4 \pi}{3}\right) + \frac{8 \pi}{3}$$
=
4*pi
$$4 \pi$$
product
4*pi 8*pi 1*----*---- 3 3
$$\frac{8 \pi}{3} \cdot 1 \cdot \frac{4 \pi}{3}$$
=
2 32*pi ------ 9
$$\frac{32 \pi^{2}}{9}$$
32*pi^2/9
Rapid solution
[src]
4*pi
x1 = ----
3
$$x_{1} = \frac{4 \pi}{3}$$
8*pi
x2 = ----
3
$$x_{2} = \frac{8 \pi}{3}$$
Numerical answer
[src]
x1 = 54.4542726622231
x2 = -192.684349420174
x3 = 20.943951023932
x4 = 58.6430628670095
x5 = -67.0206432765823
x6 = -71.2094334813686
x7 = -58.6430628670095
x8 = -79.5870138909414
x9 = -83.7758040957278
x10 = -20.943951023932
x11 = 96.342174710087
x12 = 67.0206432765823
x13 = 71.2094334813686
x14 = -92.1533845053006
x15 = -29.3215314335047
x16 = 4.18879020478639
x17 = 8.37758040957278
x18 = -117.286125734019
x19 = 33.5103216382911
x20 = 83.7758040957278
x21 = -96.342174710087
x22 = 16.7551608191456
x23 = 41.8879020478639
x24 = -41.8879020478639
x25 = -33.5103216382911
x26 = 29.3215314335047
x27 = -8.37758040957278
x28 = -4.18879020478639
x29 = -46.0766922526503
x30 = -16.7551608191456
x31 = -54.4542726622231
x32 = 79.5870138909414
x33 = 46.0766922526503
x34 = 92.1533845053006
x34 = 92.1533845053006
The graph