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Leo Authersh
Does a circular function with complex variable represent a three-dimensional graph?
For example cosiz
For example cosiz
Well, no. It represents a 90° rotation of the coordinate system.Leo Authersh said:I have read that 'i' represent the rotation of a sphere.
I have absolutely no idea of what this means.Leo Authersh said:And I have understood that similar to a two dimensional function which forms a quadratic equation, the rotation of sphere along its three dimensional axis will form a cubic equation whose roots contain complex numbers.
As I demonstrated above, the hyperbolic and circular functions are just a 90° rotation away from each other. You can combine them in different fashions, for example (assuming z=x+iy): [itex] \vert \cos(z) \vert ^{2}=\sinh(y)^{2}+\cos(x)^{2}=\cosh(y)^{2}-\sin(x)^{2}=\frac{1}{2}(\cosh(2y)+\cos(2x))[/itex]Leo Authersh said:And my question is that does a hyperbolic function that contains complex variable represent a 3-dimensional geometry in the same way a circular function represent a 2-dimensional geometry?
Can you clarify me around which axis the coordinate system is rotated 90°? Is the rotation happening alongside a different dimension than the xyz dimension?Svein said:Well, no. It represents a 90° rotation of the coordinate system.
I have absolutely no idea of what this means.
As I demonstrated above, the hyperbolic and circular functions are just a 90° rotation away from each other. You can combine them in different fashions, for example (assuming z=x+iy): [itex] \vert \cos(z) \vert ^{2}=\sinh(y)^{2}+\cos(x)^{2}=\cosh(y)^{2}-\sin(x)^{2}=\frac{1}{2}(\cosh(2y)+\cos(2x))[/itex]
Forget the "xyz dimension". The complex plane is a plane, with the real axis corresponding to the "x-axis" and the imaginary axis corresponding to the "y-axis". As you know, it is no problem to rotate the real "xy-plane" 90° without messing around with any third axis. You can describe it as x→y; y→-x or use a rotation matrix: [itex]Leo Authersh said:Can you clarify me around which axis the coordinate system is rotated 90°? Is the rotation happening alongside a different dimension than the xyz dimension?
Circular functions are mathematical functions that are defined in terms of the unit circle. They are commonly used to describe the relationships between angles and sides in a right triangle.
The most common circular functions are sine, cosine, and tangent. These functions are commonly abbreviated as sin, cos, and tan, respectively.
The unit circle is a circle with a radius of 1, centered at the origin on a coordinate plane. It is used to define the values of sine and cosine for any given angle. The x-coordinate of a point on the unit circle is equal to the cosine of the corresponding angle, while the y-coordinate is equal to the sine of the angle.
Sine and cosine are both circular functions, but they differ in the way they relate angles to the sides of a right triangle. Sine is equal to the ratio of the length of the side opposite an angle to the length of the hypotenuse, while cosine is equal to the ratio of the length of the adjacent side to the length of the hypotenuse.
Circular functions are used in many real-life applications, such as engineering, physics, and astronomy. They are also used in music and sound engineering to describe the vibrations of sound waves. Additionally, circular functions are used in navigation and mapping to calculate distances and angles between different points on the Earth's surface.