Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation x^2-x-7=0
-
Equation x^2+x-9=0
-
Equation -2(-2-y)-y-2=0
-
Equation x+5=13
- Express {x} in terms of y where:
- -6*x-13*y=17
- -7*x+5*y=1
- -6*x-19*y=7
- -19*x-4*y=-19
-
Identical expressions
- sin(four *x+p/ three)= zero
- sinus of (4 multiply by x plus p divide by 3) equally 0
- sinus of (four multiply by x plus p divide by three) equally zero
- sin(4x+p/3)=0
- sin4x+p/3=0
- sin(4*x+p/3)=O
- sin(4*x+p divide by 3)=0
-
Similar expressions
- sin(4*x-p/3)=0
sin(4*x+p/3)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$\sin{\left(\frac{p}{3} + 4 x \right)} = 0$$
- this is the simplest trigonometric equation
We get:
$$\sin{\left(\frac{p}{3} + 4 x \right)} = 0$$
This equation is transformed to
$$\frac{p}{3} + 4 x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{p}{3} + 4 x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{p}{3} + 4 x = 2 \pi n$$
$$\frac{p}{3} + 4 x = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{p}{3}$$
to right part of the equation
with the opposite sign, in total:
$$4 x = 2 \pi n - \frac{p}{3}$$
$$4 x = 2 \pi n - \frac{p}{3} + \pi$$
Divide both parts of the equation by
$$4$$
we get the answer:
$$x_{1} = \frac{\pi n}{2} - \frac{p}{12}$$
$$x_{2} = \frac{\pi n}{2} - \frac{p}{12} + \frac{\pi}{4}$$
$$\sin{\left(\frac{p}{3} + 4 x \right)} = 0$$
- this is the simplest trigonometric equation
with the change of sign in 0
We get:
$$\sin{\left(\frac{p}{3} + 4 x \right)} = 0$$
This equation is transformed to
$$\frac{p}{3} + 4 x = 2 \pi n + \operatorname{asin}{\left(0 \right)}$$
$$\frac{p}{3} + 4 x = 2 \pi n - \operatorname{asin}{\left(0 \right)} + \pi$$
Or
$$\frac{p}{3} + 4 x = 2 \pi n$$
$$\frac{p}{3} + 4 x = 2 \pi n + \pi$$
, where n - is a integer
Move
$$\frac{p}{3}$$
to right part of the equation
with the opposite sign, in total:
$$4 x = 2 \pi n - \frac{p}{3}$$
$$4 x = 2 \pi n - \frac{p}{3} + \pi$$
Divide both parts of the equation by
$$4$$
we get the answer:
$$x_{1} = \frac{\pi n}{2} - \frac{p}{12}$$
$$x_{2} = \frac{\pi n}{2} - \frac{p}{12} + \frac{\pi}{4}$$
The graph
Rapid solution
[src]
re(p) I*im(p)
x1 = - ----- - -------
12 12
$$x_{1} = - \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12}$$
re(p) pi I*im(p)
x2 = - ----- + -- - -------
12 4 12
$$x_{2} = - \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12} + \frac{\pi}{4}$$
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
$$\left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12}\right) + \left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12} + \frac{\pi}{4}\right)$$
=
re(p) pi I*im(p)
- ----- + -- - -------
6 4 6
$$- \frac{\operatorname{re}{\left(p\right)}}{6} - \frac{i \operatorname{im}{\left(p\right)}}{6} + \frac{\pi}{4}$$
product
/ re(p) I*im(p)\ / re(p) pi I*im(p)\ |- ----- - -------|*|- ----- + -- - -------| \ 12 12 / \ 12 4 12 /
$$\left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12}\right) \left(- \frac{\operatorname{re}{\left(p\right)}}{12} - \frac{i \operatorname{im}{\left(p\right)}}{12} + \frac{\pi}{4}\right)$$
=
(I*im(p) + re(p))*(-3*pi + I*im(p) + re(p))
-------------------------------------------
144
$$\frac{\left(\operatorname{re}{\left(p\right)} + i \operatorname{im}{\left(p\right)}\right) \left(\operatorname{re}{\left(p\right)} + i \operatorname{im}{\left(p\right)} - 3 \pi\right)}{144}$$
(i*im(p) + re(p))*(-3*pi + i*im(p) + re(p))/144