Angle generated by a counter-clockwise rotation, we call angle a Positive Angle. On the other hand, if an angle is generated by a clockwise rotation we call angle a Negative Angle
A quadratic function is of form y = ax² + bx + c where a ≠ 0 and a, b, c are real number. Graph of quadratic functions are always a parabola either opening upwards or downwards.
To plot graph of any quadratic function, we need answers of these question
What is sign of ‘a’ or coefficient of x2 in quadratic function?
Whether the graph of quadratic function intersects with x-axis? And if it does at what point does it intersect?
Let z = (a + i b) be any complex number. The nth root of complex number z is given by z1/n where n → θ (i.e. set of rational numbers).
Convert the given complex number, into polar form.
Add 2kπ to the argument of the complex number converted into polar form.
Raise index 1/n to the power of z to calculate the nth root of complex number.
The complex number z in geometrical form is written as z = x + iy.In geometrical representation complex number z is represented by a point P(x, y) on the complex plane or the argand plane where OA =x is x-intecept and AP=y is y-intercept.
In polar representation a complex number z is represented by two parameters ‘r’ and ‘θ’. Parameter ‘r’ is the modulus of complex number and parameter ‘θ’ is the angle which the line OP makes with the positive direction of x-axis. It is also called argument of complex number and is denoted by arg(z).Finding the value of these two parameters from parameters x and y will help us convert the complex number to polar form.
Let a + i b be a complex number whose logarithm is to be found.
Step 1: Convert the given complex number, into polar form.
Where amplitude and argument is given.
Step 2: Use Euler’s Theorem to rewrite complex number in polar form to exponential form.
There r (cos θ + isinθ) is written as reiθ. This means that
a+ i b= reiθ
Step 3: Take logarithm of both sides we get.
Argand Plane is used to represent the complex number. The horizontal axis of Argand plane represents the real part while vertical axis represents the imaginary part of complex number.
The modulus of complex number is distance of a point P (which represents complex number in Argand Plane) from the origin....
Step 1 : Invert the number
If z = a + i b is a complex number, then reciprocal of it is given by..
Step 2: Multiply numerator and denominator by conjugate
Multiply numerator and denominator of the inverted number by conjugate of denominator...
Step 3: Simplify and find the reciprocal
Simplify above equation in step (2). Numerator is multiplied by 1 and is already simplified...