Mister Exam
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How to use it?
- Equation:
- Equation x^4-29x^2+100=0
- Equation sin(pi(x-3))/4=sqrt2/2
-
Equation 8-5(8+3x)=13
-
Equation 2x^2+13-13x=x+1
- Express {x} in terms of y where:
- 5*x+5*y=20
- 4*x-16*y=6
- sqrt(x^2+y^2+4x-6y-12)+(x^2+10x-27)=8-|y-9|+|y-3|
- 10*x-1*y=10
-
Identical expressions
- sin(pi(x- three))/ four =sqrt two /2
- sinus of ( Pi (x minus 3)) divide by 4 equally square root of 2 divide by 2
- sinus of ( Pi (x minus three)) divide by four equally square root of two divide by 2
- sin(pi(x-3))/4=√2/2
- sinpix-3/4=sqrt2/2
- sin(pi(x-3)) divide by 4=sqrt2 divide by 2
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Similar expressions
- sin((pi(x-3))/4)=sqrt(2)
- sin(pi(x+3))/4=sqrt2/2
- sin(pi*(x-3)/4)=5*pi/4
- cospi(4x+132)/4=-sqrt(2)/2
- cos(x)=sqrt(2)/2
- sin(-t/4)=-sqrt(2)/2
sin(pi(x-3))/4=sqrt2/2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
___
sin(pi*(x - 3)) \/ 2
--------------- = -----
4 2
$$\frac{\sin{\left(\pi \left(x - 3\right) \right)}}{4} = \frac{\sqrt{2}}{2}$$
Detail solution
Given the equation
$$\frac{\sin{\left(\pi \left(x - 3\right) \right)}}{4} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
The equation is transformed to
$$\sin{\left(\pi x \right)} = - 2 \sqrt{2}$$
As right part of the equation
modulo =
but sin
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
$$\frac{\sin{\left(\pi \left(x - 3\right) \right)}}{4} = \frac{\sqrt{2}}{2}$$
- this is the simplest trigonometric equation
Divide both parts of the equation by -1/4
The equation is transformed to
$$\sin{\left(\pi x \right)} = - 2 \sqrt{2}$$
As right part of the equation
modulo =
True
but sin
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
Sum and product of roots
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sum
/ / ___\\ / / ___\\ / / ___\\ / / ___\\
pi + re\asin\2*\/ 2 // I*im\asin\2*\/ 2 // re\asin\2*\/ 2 // I*im\asin\2*\/ 2 //
---------------------- + ------------------- + - ----------------- - -------------------
pi pi pi pi
$$\left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi}\right) + \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi}\right)$$
=
/ / ___\\ / / ___\\
pi + re\asin\2*\/ 2 // re\asin\2*\/ 2 //
---------------------- - -----------------
pi pi
$$- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi} + \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)} + \pi}{\pi}$$
product
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\ |pi + re\asin\2*\/ 2 // I*im\asin\2*\/ 2 //| | re\asin\2*\/ 2 // I*im\asin\2*\/ 2 //| |---------------------- + -------------------|*|- ----------------- - -------------------| \ pi pi / \ pi pi /
$$\left(\frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi}\right) \left(- \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi}\right)$$
=
/ / / ___\\ / / ___\\\ / / / ___\\ / / ___\\\
-\I*im\asin\2*\/ 2 // + re\asin\2*\/ 2 ///*\pi + I*im\asin\2*\/ 2 // + re\asin\2*\/ 2 ///
------------------------------------------------------------------------------------------
2
pi
$$- \frac{\left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)} + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}\right) \left(\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)} + \pi + i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}\right)}{\pi^{2}}$$
-(i*im(asin(2*sqrt(2))) + re(asin(2*sqrt(2))))*(pi + i*im(asin(2*sqrt(2))) + re(asin(2*sqrt(2))))/pi^2
Rapid solution
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/ / ___\\ / / ___\\
pi + re\asin\2*\/ 2 // I*im\asin\2*\/ 2 //
x1 = ---------------------- + -------------------
pi pi
$$x_{1} = \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)} + \pi}{\pi} + \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi}$$
/ / ___\\ / / ___\\
re\asin\2*\/ 2 // I*im\asin\2*\/ 2 //
x2 = - ----------------- - -------------------
pi pi
$$x_{2} = - \frac{\operatorname{re}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi} - \frac{i \operatorname{im}{\left(\operatorname{asin}{\left(2 \sqrt{2} \right)}\right)}}{\pi}$$
x2 = -re(asin(2*sqrt(2)))/pi - i*im(asin(2*sqrt(2)))/pi
Numerical answer
[src]
x1 = 1.5 - 0.541140241435849*i
x2 = -0.5 + 0.541140241435849*i
x2 = -0.5 + 0.541140241435849*i