Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation 3x+3=-2-7x
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Equation x+5=12
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Equation cos(x)=-3
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Equation 3x-1=2x
- Express {x} in terms of y where:
- -3*x-5*y=0
- -15*x+13*y=2
- -14*x-3*y=16
- -17*x-9*y=0
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Identical expressions
- 4cosz+ five = zero
- 4 co sinus of e of z plus 5 equally 0
- 4 co sinus of e of z plus five equally zero
- 4cosz+5=O
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Similar expressions
- 4cosz-5=0
4cosz+5=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the equation
$$4 \cos{\left(z \right)} + 5 = 0$$
- this is the simplest trigonometric equation
We get:
$$4 \cos{\left(z \right)} = -5$$
The equation is transformed to
$$\cos{\left(z \right)} = - \frac{5}{4}$$
As right part of the equation
modulo =
but cos
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
$$4 \cos{\left(z \right)} + 5 = 0$$
- this is the simplest trigonometric equation
Move 5 to right part of the equation
with the change of sign in 5
We get:
$$4 \cos{\left(z \right)} = -5$$
Divide both parts of the equation by 4
The equation is transformed to
$$\cos{\left(z \right)} = - \frac{5}{4}$$
As right part of the equation
modulo =
True
but cos
can no be more than 1 or less than -1
so the solution of the equation d'not exist.
Rapid solution
[src]
z1 = -re(acos(-5/4)) + 2*pi - I*im(acos(-5/4))
$$z_{1} = - \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}$$
z2 = I*im(acos(-5/4)) + re(acos(-5/4))
$$z_{2} = \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}$$
z2 = re(acos(-5/4)) + i*im(acos(-5/4))
Sum and product of roots
[src]
sum
-re(acos(-5/4)) + 2*pi - I*im(acos(-5/4)) + I*im(acos(-5/4)) + re(acos(-5/4))
$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}\right) + \left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}\right)$$
=
2*pi
$$2 \pi$$
product
(-re(acos(-5/4)) + 2*pi - I*im(acos(-5/4)))*(I*im(acos(-5/4)) + re(acos(-5/4)))
$$\left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}\right) \left(- \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + 2 \pi - i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}\right)$$
=
-(I*im(acos(-5/4)) + re(acos(-5/4)))*(-2*pi + I*im(acos(-5/4)) + re(acos(-5/4)))
$$- \left(\operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}\right) \left(- 2 \pi + \operatorname{re}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)} + i \operatorname{im}{\left(\operatorname{acos}{\left(- \frac{5}{4} \right)}\right)}\right)$$
-(i*im(acos(-5/4)) + re(acos(-5/4)))*(-2*pi + i*im(acos(-5/4)) + re(acos(-5/4)))
Numerical answer
[src]
z1 = 3.14159265358979 + 0.693147180559945*i
z2 = 3.14159265358979 - 0.693147180559945*i
z2 = 3.14159265358979 - 0.693147180559945*i