Identities/Formulae Flashcards
cos^2x + sin^2x
= 1
sin2x
= 2sinxcosx
cos2x
= cos^2x - sin^2x / 2cos^2x -1 / 1 - 2sin^2x
sec^2x
= 1 + tan^2x
cosec^2x
= 1 + cot^2x
sin^2x
= 0.5 - 0.5cos2x
cos^2x
= 0.5 + 0.5cos2x
if det(M) = 0
M is Singular, ad - bc = 0
How many inverses do Singular Matrices have?
No inverses as they have no determinant so 1 / det(M) is undefined so there is no working matrix.
sinhx
= 0.5(e^x - e^-x)
coshx
= 0.5(e^x + e^-x)
arsinhx
= ln(x + root(x^2 +1))
arcoshx
= ln(x + root(x^2 -1))
artanhx
= 0.5ln((1 + x)/ (1 - x))
How do Hyperbolic identites differ from standard?
Wherever a hyperbolic function has a product of two sines, the product of the hyperbolic sines must be negated.
cosh^2x - sinh^2x
= 1
1 - tanh^2x
= sech^2x
coth^2x - 1
= cosech^2x
z^n
r^n(cosnx + isinnx)
Volume of Rev with a Parametric Equation
Pi (int (y^2 (dx/dt) dt))
Perpendicular Tangent
x = rcosx
Parallel Tangent
y = rsinx
(z + z^-1)
2cosx
(z^n + z^-n)
2cosnx
(z - z^-1)
2isinx
(z^n - z^-n)
2isinnx
How to solve a Partial Fraction with a Quadratic Factor in the denominator?
This cannot be solved normally, there will not be a quadratic term on the RHS of the equation so therefore one of the terms will always be wrong. So therefore in order to solve this, we must include a term one below the term in the denominator, which for a quadratic, is a linear term such as Bx + C. If this was a cubic on the denominator then we would have to include a quadratic term on the numerator etc. And then this can be solved normally.
For example:
(5x - 7) / (x +1) (x^2 + 1) = A / (x +1) + (Bx + C) / (x^2 + 1)
How to find the inverse of a 3x3 matrix?
Form the new matrix by finding the determinant for each term within the matrix. Then apply the new signs to each of the terms, pluses in a diagonal shape, and then minuses elsewhere. Then take the transpose, switching each term in diagonal lines from right to left. Finally, find the original determinant of the matrix and place 1 / det(M) outside the inverse matrix you just formed.
