Mister Exam
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How to use it?
- Equation:
-
Equation -x^2-4*x-3=0
-
Equation 3x+3=-2-7x
-
Equation sin(-x)=0
-
Equation (x+3)^3=81*(x+3)
- Express {x} in terms of y where:
- -5*x+9*y=0
- 16*x+2*y=14
- 18*x-1*y=6
- 19*x-9*y=8
- Derivative of:
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1-cos^2x
-
2sinx
- Graphing y =:
-
2sinx
- Integral of d{x}:
-
2sinx
-
Identical expressions
- one -cos^2x=2sinx
- 1 minus co sinus of e of squared x equally 2 sinus of x
- one minus co sinus of e of squared x equally 2 sinus of x
- 1-cos2x=2sinx
- 1-cos²x=2sinx
- 1-cos to the power of 2x=2sinx
-
Similar expressions
- 2*sin(x)-3=0
- 1+cos^2x=2sinx
- 2*sin(x)=-1
- -2*sin(x)*cos(x)=1/2
1-cos^2x=2sinx equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
2 1 - cos (x) = 2*sin(x)
$$1 - \cos^{2}{\left(x \right)} = 2 \sin{\left(x \right)}$$
Detail solution
Given the equation
$$1 - \cos^{2}{\left(x \right)} = 2 \sin{\left(x \right)}$$
transform
$$\left(\sin{\left(x \right)} - 2\right) \sin{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -2$$
$$c = 0$$
, then
Because D > 0, then the equation has two roots.
or
$$w_{1} = 2$$
$$w_{2} = 0$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
$$1 - \cos^{2}{\left(x \right)} = 2 \sin{\left(x \right)}$$
transform
$$\left(\sin{\left(x \right)} - 2\right) \sin{\left(x \right)} = 0$$
$$\sin^{2}{\left(x \right)} - 2 \sin{\left(x \right)} = 0$$
Do replacement
$$w = \sin{\left(x \right)}$$
This equation is of the form
a*w^2 + b*w + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$w_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$w_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -2$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(-2)^2 - 4 * (1) * (0) = 4
Because D > 0, then the equation has two roots.
w1 = (-b + sqrt(D)) / (2*a)
w2 = (-b - sqrt(D)) / (2*a)
or
$$w_{1} = 2$$
$$w_{2} = 0$$
do backward replacement
$$\sin{\left(x \right)} = w$$
Given the equation
$$\sin{\left(x \right)} = w$$
- this is the simplest trigonometric equation
This equation is transformed to
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
Or
$$x = 2 \pi n + \operatorname{asin}{\left(w \right)}$$
$$x = 2 \pi n - \operatorname{asin}{\left(w \right)} + \pi$$
, where n - is a integer
substitute w:
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(w_{1} \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{1} = 2 \pi n + \operatorname{asin}{\left(2 \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(w_{2} \right)}$$
$$x_{2} = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$x_{2} = 2 \pi n$$
$$x_{3} = 2 \pi n - \operatorname{asin}{\left(w_{1} \right)} + \pi$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{3} = 2 \pi n + \pi - \operatorname{asin}{\left(2 \right)}$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(w_{2} \right)} + \pi$$
$$x_{4} = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
$$x_{4} = 2 \pi n + \pi$$
Sum and product of roots
[src]
sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 ||
2*re|atan|- - -------|| + 2*I*im|atan|- - -------|| + 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
\ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$\left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) + \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 || | |1 I*\/ 3 ||
2*re|atan|- + -------|| + 2*re|atan|- - -------|| + 2*I*im|atan|- + -------|| + 2*I*im|atan|- - -------||
\ \2 2 // \ \2 2 // \ \2 2 // \ \2 2 //
$$2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ | | |1 I*\/ 3 || | |1 I*\/ 3 ||| | | |1 I*\/ 3 || | |1 I*\/ 3 ||| 0*|2*re|atan|- - -------|| + 2*I*im|atan|- - -------|||*|2*re|atan|- + -------|| + 2*I*im|atan|- + -------||| \ \ \2 2 // \ \2 2 /// \ \ \2 2 // \ \2 2 ///
$$0 \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}\right) \left(2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}\right)$$
=
0
$$0$$
0
Rapid solution
[src]
x1 = 0
$$x_{1} = 0$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x2 = 2*re|atan|- - -------|| + 2*I*im|atan|- - -------||
\ \2 2 // \ \2 2 //
$$x_{2} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} - \frac{\sqrt{3} i}{2} \right)}\right)}$$
/ / ___\\ / / ___\\
| |1 I*\/ 3 || | |1 I*\/ 3 ||
x3 = 2*re|atan|- + -------|| + 2*I*im|atan|- + -------||
\ \2 2 // \ \2 2 //
$$x_{3} = 2 \operatorname{re}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)} + 2 i \operatorname{im}{\left(\operatorname{atan}{\left(\frac{1}{2} + \frac{\sqrt{3} i}{2} \right)}\right)}$$
x3 = 2*re(atan(1/2 + sqrt(3)*i/2)) + 2*i*im(atan(1/2 + sqrt(3)*i/2))
Numerical answer
[src]
x1 = 31.4159265358979
x2 = 3.14159265358979
x3 = -47.1238898038469
x4 = -12.5663706143592
x5 = -34.5575191894877
x6 = -69.1150383789755
x7 = 75.398223686155
x8 = -65.9734457253857
x9 = -50.2654824574367
x10 = -56.5486677646163
x11 = 59.6902604182061
x12 = 72.2566310325652
x13 = 91.106186954104
x14 = -91.106186954104
x15 = -62.8318530717959
x16 = -6.28318530717959
x17 = -232.477856365645
x18 = 62.8318530717959
x19 = 6.28318530717959
x20 = -25.1327412287183
x21 = 94.2477796076938
x22 = -9.42477796076938
x23 = -37.6991118430775
x24 = 65.9734457253857
x25 = -100.530964914873
x26 = -43.9822971502571
x27 = 25.1327412287183
x28 = 21.9911485751286
x29 = 87.9645943005142
x30 = -40.8407044966673
x31 = -97.3893722612836
x32 = 43.9822971502571
x33 = -53.4070751110265
x34 = 97.3893722612836
x35 = 100.530964914873
x36 = -94.2477796076938
x37 = -31.4159265358979
x38 = 18.8495559215388
x39 = 78.5398163397448
x40 = -18.8495559215388
x41 = 53.4070751110265
x42 = 47.1238898038469
x43 = 12.5663706143592
x44 = 81.6814089933346
x45 = 34.5575191894877
x46 = -75.398223686155
x47 = -15.707963267949
x48 = 50.2654824574367
x49 = -81.6814089933346
x50 = -3.14159265358979
x51 = -59.6902604182061
x52 = -28.2743338823081
x53 = -87.9645943005142
x54 = 9.42477796076938
x55 = -21.9911485751286
x56 = 56.5486677646163
x57 = -185.353966561798
x58 = 15.707963267949
x59 = 84.8230016469244
x60 = -78.5398163397448
x61 = 37.6991118430775
x62 = -72.2566310325652
x63 = -84.8230016469244
x64 = 69.1150383789755
x65 = 0.0
x66 = 128.805298797182
x67 = 28.2743338823081
x68 = 40.8407044966673
x68 = 40.8407044966673