Mister Exam

Other calculators

sin(4*x+p/3)=0 equation

The teacher will be very surprised to see your correct solution 😉

v

Numerical solution:

Do search numerical solution at [, ]

The solution

You have entered [src] 
   /      p\    
sin|4*x + -| = 0
   \      3/
sin⁡(p3+4x)=0\sin{\left(\frac{p}{3} + 4 x \right)} = 0
Simplify
Detail solution
Given the equation
sin⁡(p3+4x)=0\sin{\left(\frac{p}{3} + 4 x \right)} = 0
- this is the simplest trigonometric equation
with the change of sign in 0

We get:
sin⁡(p3+4x)=0\sin{\left(\frac{p}{3} + 4 x \right)} = 0
This equation is transformed to
p3+4x=2πn+asin⁡(0)\frac{p}{3} + 4 x = 2 \pi n + \operatorname{asin}{\left(0 \right)}
p3+4x=2πn−asin⁡(0)+π\frac{p}{3} + 4 x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi
Or
p3+4x=2πn\frac{p}{3} + 4 x = 2 \pi n
p3+4x=2πn+π\frac{p}{3} + 4 x = 2 \pi n + \pi
, where n - is a integer
Move
p3\frac{p}{3}
to right part of the equation
with the opposite sign, in total:
4x=2πn−p34 x = 2 \pi n - \frac{p}{3}
4x=2πn−p3+π4 x = 2 \pi n - \frac{p}{3} + \pi
Divide both parts of the equation by
44
we get the answer:
x1=πn2−p12x_{1} = \frac{\pi n}{2} - \frac{p}{12}
x2=πn2−p12+π4x_{2} = \frac{\pi n}{2} - \frac{p}{12} + \frac{\pi}{4}
The graph
Rapid solution [src] 
       re(p)   I*im(p)
x1 = - ----- - -------
         12       12
x1=−re⁡(p)12−iim⁡(p)12x_{1} = - \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12} 
       re(p)   pi   I*im(p)
x2 = - ----- + -- - -------
         12    4       12
x2=−re⁡(p)12−iim⁡(p)12+π4x_{2} = - \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12} + \frac{\pi}{4} 
x2 = -re(p)/12 - i*im(p)/12 + pi/4
Sum and product of roots [src] 
sum
  re(p)   I*im(p)     re(p)   pi   I*im(p)
- ----- - ------- + - ----- + -- - -------
    12       12         12    4       12
(−re⁡(p)12−iim⁡(p)12)+(−re⁡(p)12−iim⁡(p)12+π4)\left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12}\right) + \left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12} + \frac{\pi}{4}\right) 
=
  re(p)   pi   I*im(p)
- ----- + -- - -------
    6     4       6
−re⁡(p)6−iim⁡(p)6+π4- \frac{\operatorname{re}{\left(p\right)}}{6} - \frac{i \operatorname{im}{\left(p\right)}}{6} + \frac{\pi}{4} 
product
/  re(p)   I*im(p)\ /  re(p)   pi   I*im(p)\
|- ----- - -------|*|- ----- + -- - -------|
\    12       12  / \    12    4       12  /
(−re⁡(p)12−iim⁡(p)12)(−re⁡(p)12−iim⁡(p)12+π4)\left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12}\right) \left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12} + \frac{\pi}{4}\right) 
=
(I*im(p) + re(p))*(-3*pi + I*im(p) + re(p))
-------------------------------------------
                    144
(re⁡(p)+iim⁡(p))(re⁡(p)+iim⁡(p)−3π)144\frac{\left(\operatorname{re}{\left(p\right)} + i \operatorname{im}{\left(p\right)}\right) \left(\operatorname{re}{\left(p\right)} + i \operatorname{im}{\left(p\right)} - 3 \pi\right)}{144} 
(i*im(p) + re(p))*(-3*pi + i*im(p) + re(p))/144
    © Mister Exam – Calculator