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Hyperbolic Functions
Sinh, cosh, tanh are hyperbolic functions.
Sinh x = 0.5(e^x-e^-x)
Cosh x = 0.5(e^x+e^-x)
Tanh x = (e^2x-1)/(e^2x+1)
To integrate hyperbolic functions, put them in exponential form or assume multiple of 1 and integrate by parts.
Differentiation of inverse hyperbolic functions (arsinh, arcosh, artanh) is referenced in the formula booklet.
Differential of cosh = sinh. Differential of sinh = -cosh. This is opposite to normal trigonometrical differentiatials.
Matrices
A 3×3 matrix is a matrix of the form: (a,b,c|d,e,f|g,h,i)
The determinant of a 3×3 matrix can be found by Sarrus’ Method: (aei+bfg+cdh)-(ceg+fha+ibd)
To find the inverse M^-1 of a 3×3 matrix M, a number of steps have to be followed:
- Find the determinant
- Find the cofactors of every cell in the matrix (This means the determinant of the 2×2 matrix left after eliminating the row and column of the cell of which the cofactor is being found)
- Alter the signs of the cofactors according to the matrix (+,-,+|-,+,-|+,-,+)
- Transpose the matrix so that each row becomes a column and vice versa, so (a,b,c|d,e,f|g,h,i) becomes (a,d,g|b,e,h|c,f,i).
- Divide the resultant matrix by the determinant of the original matrix
The matrix M-(lamda)I = (a-lamba,b,c|d,e-lamda,f|g,h,i-lamda). The characteristic equation of a matrix is equating the determinant to 0, ending up with an equation in terms of lamda.
Eigenvalues are values of lamda for which det(M-(lamda)I)=0.
To obtain eigenvectors, first obtain simultaneous equations as follows:
ax+by+cz=(lamda)x
dx+ey+fz=(lamda)y
gx+hy+iz=(lamda)z
These can be solved to find the relationship between x, y, and z, and this can be put into the form of a 3×1 eigenvector.
To verify if a matrix is an eigenvector of a matrix, postmultiply the matrix by the eigenvector. If the output is a scalar multiple of the eigenvector, then it is valid.
Once eigenvalues and eigenvectors are known, it is possible to combine them together like so:
Eigenvectors (a,b,c) (d,e,f) (g,h,i)
Eigenvalues j, k, l respectively
P = (a,d,g|b,e,h|c,f,i)
D = (j,0,0|0,k,0|0,0,l)
Now the matrix M can be written as PDP^-1. This is useful for finding powers of M as M^n = PD^nP^-1 and D is a diagonal matrix so D^n = (j^n,0,0|0,k^n,0|0,0,l^n)
Cayley Hamilton theorem states that every square matrix satisfies its own characteristic equation.
Complex Numbers
A complex number is a number of the form a+bj where j is sqrt(-1).
It can be written as a+bj (Real and Imaginary parts), as c(cos d + j sin d) (Where c is the modulus and d is the argument, c cos d is the real part and c sin d is the imaginary part), or as ce^(jd).
cos theta + j sin theta can be written as e^(j(theta)).
An argand diagram is a graph of real part (x-axis) against imaginary part (y-axis).
The modulus of an imaginary number is its distance from the origin to its point on an argand diagram, and can be found by sqrt(a^2+b^2) or c.
The argument of a complex number is the angle that it makes, in radians, with the x axis. Real numbers have an argument of 0 as they all lie on the x axis. The argument can be found by arctan(b/a) or d.
If w is a complex number a+bj then w* is the complex conjugate a-bj.
j^2 = -1
(cos theta + j sin theta)^n = cos n(theta) + j sin n(theta). This is known as De Moivre’s theorem.
To find the roots of a complex number of the form a+bj, the modulus and argument must be worked out. There are as many solutions as the root number (eg. to find the nth roots of a+bj then there are n answers). The modulus of all the roots is the same and is the nth root of the modulus of the complex number. The argument of the first root can be found by taking the argument of the complex number and dividing it by n. Then to find each of the other arguments for the other roots, add (2pi/n) to the argument each time.
If given two series where:
C = a cos theta + b cos 2theta + c cos 3theta …
S = d sin theta + e sin 2theta + f sin 3theta …
To find another equation for C or S, first multiply S by j and then add C, so that you end up with an infinite series C+jS. Then group together common values of cos and sin theta and use De Moivre’s theorem to group them together into a single series. This series should be recognisable as being arithmetic or geometric in the formula booklet, which can then be used to sum the infinite series.
MacLaurin Series
A MacLaurin series is a series of a function to infinity expressed in coefficients of powers of x.
The MacLaurin series can be written as: f(x) = f(0) + (1/1!)f’(0)x + (1/2!)f”(0)x^2 + (1/3!)f”’(0)x^3 + …
A MacLaurin series is only generally valid for small values of x.
Integration
The formula book contains a good reference for integration of various trigonometrical functions and inverse trigonometrical functions. The integral must be put into a correct form in the integration, ensuring that the coefficient of x^2 on the bottom is always 1.
Alternatively a trigonometrical substitution could be used.
If an integral involves sqrt(a^2-x^2) then let x = a sin theta and use (1-sin^2(theta))=cos^2(theta)
If an integral involves sqrt(a^2+x^2) then let x = a tan theta and use (1+tan^2(theta))=sec^2(theta)
If an integral involves sqrt(x^2-a^2) then let x = a sec theta and use (sec^2(theta)-1)=tan^2(theta)
Polar Coordinates
Polar Coordinates can be drawn using a graphical calculator, and are in the form r=f(theta). Any point on the line for which r is negative should be drawn using a dotted line. r is the distance between the point on the curve and the origin, and theta is the angle that it makes against the horizontal in radians
The area of a point of a polar coordinate is defined as the integral between the limits of theta=b and theta=a of (1/2)r^2. Values of theta should be relevant and in radians.
Yeah, that’s about all that’s needed to know about Polar Coordinates. Only worth about 7-8 marks on each paper, so there you go. ASCII polar coordinates for the win.
