Mister Exam
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- Equation:
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Equation 3x^2-5x-2=0
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Equation 2(7x-7)=7(2x-3)+7
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Equation sqrt(x^3-4*x^2-10*x+29)=3-x
- Express {x} in terms of y where:
- -6*x-2*y=-10
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- -8*x-12*y=15
- sin(xy)+cos(xy)=tg(x+y)
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Identical expressions
- (one -tg(x/ two))/(one -ctg(x/ two))= two sin(x/2)
- (1 minus tg(x divide by 2)) divide by (1 minus ctg(x divide by 2)) equally 2 sinus of (x divide by 2)
- (one minus tg(x divide by two)) divide by (one minus ctg(x divide by two)) equally two sinus of (x divide by 2)
- 1-tgx/2/1-ctgx/2=2sinx/2
- (1-tg(x divide by 2)) divide by (1-ctg(x divide by 2))=2sin(x divide by 2)
-
Similar expressions
- (log2(sin(x))^2+log2(sin(x)))/2cos(x)+sqrt3=0
- (1-tg(x/2))/(1+ctg(x/2))=2sin(x/2)
- 2*sin(x)/2=1-cos(x)
- 2sin*x/2=√3
- (1+tg(x/2))/(1-ctg(x/2))=2sin(x/2)
(1-tg(x/2))/(1-ctg(x/2))=2sin(x/2) equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
/x\
1 - tan|-|
\2/ /x\
---------- = 2*sin|-|
/x\ \2/
1 - cot|-|
\2/
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{1 - \cot{\left(\frac{x}{2} \right)}} = 2 \sin{\left(\frac{x}{2} \right)}$$
Detail solution
Given the equation
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{1 - \cot{\left(\frac{x}{2} \right)}} = 2 \sin{\left(\frac{x}{2} \right)}$$
transform
$$\frac{- 2 \sqrt{2} \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + \tan{\left(\frac{x}{2} \right)} - 1}{\cot{\left(\frac{x}{2} \right)} - 1} = 0$$
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{1 - \cot{\left(\frac{x}{2} \right)}} - 2 \sin{\left(\frac{x}{2} \right)} = 0$$
Do replacement
$$w = \cot{\left(\frac{x}{2} \right)}$$
Given the equation:
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{1 - \cot{\left(\frac{x}{2} \right)}} - 2 \sin{\left(\frac{x}{2} \right)} = 0$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case
so we get the equation
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}} = 2 \left(1 - \cot{\left(\frac{x}{2} \right)}\right)$$
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}} = 2 - 2 \cot{\left(\frac{x}{2} \right)}$$
Expand brackets in the left part
Expand brackets in the right part
This equation has no roots
do backward replacement
$$\cot{\left(\frac{x}{2} \right)} = w$$
substitute w:
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{1 - \cot{\left(\frac{x}{2} \right)}} = 2 \sin{\left(\frac{x}{2} \right)}$$
transform
$$\frac{- 2 \sqrt{2} \cos{\left(\frac{x}{2} + \frac{\pi}{4} \right)} + \tan{\left(\frac{x}{2} \right)} - 1}{\cot{\left(\frac{x}{2} \right)} - 1} = 0$$
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{1 - \cot{\left(\frac{x}{2} \right)}} - 2 \sin{\left(\frac{x}{2} \right)} = 0$$
Do replacement
$$w = \cot{\left(\frac{x}{2} \right)}$$
Given the equation:
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{1 - \cot{\left(\frac{x}{2} \right)}} - 2 \sin{\left(\frac{x}{2} \right)} = 0$$
Use proportions rule:
From a1/b1 = a2/b2 should a1*b2 = a2*b1,
In this case
a1 = 1 - tan(x/2)
b1 = 1 - cot(x/2)
a2 = 2
b2 = 1/sin(x/2)
so we get the equation
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}} = 2 \left(1 - \cot{\left(\frac{x}{2} \right)}\right)$$
$$\frac{1 - \tan{\left(\frac{x}{2} \right)}}{\sin{\left(\frac{x}{2} \right)}} = 2 - 2 \cot{\left(\frac{x}{2} \right)}$$
Expand brackets in the left part
1+tan+x/2)/sinx/2 = 2 - 2*cot(x/2)
Expand brackets in the right part
1+tan+x/2)/sinx/2 = 2 - 2*cotx/2
This equation has no roots
do backward replacement
$$\cot{\left(\frac{x}{2} \right)} = w$$
substitute w:
Rapid solution
[src]
-8*pi
x1 = -----
3
$$x_{1} = - \frac{8 \pi}{3}$$
-4*pi
x2 = -----
3
$$x_{2} = - \frac{4 \pi}{3}$$
4*pi
x3 = ----
3
$$x_{3} = \frac{4 \pi}{3}$$
8*pi
x4 = ----
3
$$x_{4} = \frac{8 \pi}{3}$$
x4 = 8*pi/3
Sum and product of roots
[src]
sum
8*pi 4*pi 4*pi 8*pi - ---- - ---- + ---- + ---- 3 3 3 3
$$\left(\left(- \frac{8 \pi}{3} - \frac{4 \pi}{3}\right) + \frac{4 \pi}{3}\right) + \frac{8 \pi}{3}$$
=
0
$$0$$
product
-8*pi -4*pi 4*pi 8*pi -----*-----*----*---- 3 3 3 3
$$\frac{8 \pi}{3} \frac{4 \pi}{3} \cdot - \frac{8 \pi}{3} \left(- \frac{4 \pi}{3}\right)$$
=
4 1024*pi -------- 81
$$\frac{1024 \pi^{4}}{81}$$
1024*pi^4/81
Numerical answer
[src]
x1 = -69.1150383789755
x2 = -100.530964914873
x3 = 33.5103216382911
x4 = -25.1327412287183
x5 = -75.398223686155
x6 = -33.5103216382911
x7 = -50.2654824574367
x8 = -46.0766922526503
x9 = 81.6814089933346
x10 = 50.2654824574367
x11 = -43.9822971502571
x12 = -37.6991118430775
x13 = -16.7551608191456
x14 = 108.908545324446
x15 = 25.1327412287183
x16 = -18.8495559215388
x17 = -8.37758040957278
x18 = -92.1533845053006
x19 = -96.342174710087
x20 = 31.4159265358979
x21 = -56.5486677646163
x22 = 18.8495559215388
x23 = -232.477856365645
x24 = -6.28318530717959
x25 = -62.8318530717959
x26 = -29.3215314335047
x27 = 12.5663706143592
x28 = -54.4542726622231
x29 = 69.1150383789755
x30 = 58.6430628670095
x31 = 56.5486677646163
x32 = 6.28318530717959
x33 = 83.7758040957278
x34 = 37.6991118430775
x35 = -83.7758040957278
x36 = 96.342174710087
x37 = 92.1533845053006
x38 = -31.4159265358979
x39 = 41.8879020478639
x40 = 100.530964914873
x41 = 71.2094334813686
x42 = 94.2477796076938
x43 = 16.7551608191456
x44 = -12.5663706143592
x45 = 54.4542726622231
x46 = -79.5870138909414
x47 = -94.2477796076938
x48 = 67.0206432765823
x49 = 43.9822971502571
x50 = 75.398223686155
x51 = -41.8879020478639
x52 = 62.8318530717959
x53 = 87.9645943005142
x54 = -4.18879020478639
x55 = -20.943951023932
x56 = -81.6814089933346
x57 = -87.9645943005142
x58 = -71.2094334813686
x59 = 4.18879020478639
x60 = 46.0766922526503
x61 = 8.37758040957278
x61 = 8.37758040957278