Identities/Formulae

Question 1

cos^2x + sin^2x

Answer 1

= 1

Question 2

sin2x

Answer 2

= 2sinxcosx

Question 3

cos2x

Answer 3

= cos^2x - sin^2x / 2cos^2x -1 / 1 - 2sin^2x

Question 4

sec^2x

Answer 4

= 1 + tan^2x

Question 5

cosec^2x

Answer 5

= 1 + cot^2x

Question 6

sin^2x

Answer 6

= 0.5 - 0.5cos2x

Question 7

cos^2x

Answer 7

= 0.5 + 0.5cos2x

Question 8

if det(M) = 0

Answer 8

M is Singular, ad - bc = 0

Question 9

How many inverses do Singular Matrices have?

Answer 9

No inverses as they have no determinant so 1 / det(M) is undefined so there is no working matrix.

Question 10

sinhx

Answer 10

= 0.5(e^x - e^-x)

Question 11

coshx

Answer 11

= 0.5(e^x + e^-x)

Question 12

arsinhx

Answer 12

= ln(x + root(x^2 +1))

Question 13

arcoshx

Answer 13

= ln(x + root(x^2 -1))

Question 14

artanhx

Answer 14

= 0.5ln((1 + x)/ (1 - x))

Question 15

How do Hyperbolic identites differ from standard?

Answer 15

Wherever a hyperbolic function has a product of two sines, the product of the hyperbolic sines must be negated.

Question 16

cosh^2x - sinh^2x

Answer 16

= 1

Question 17

1 - tanh^2x

Answer 17

= sech^2x

Question 18

coth^2x - 1

Answer 18

= cosech^2x

Question 19

z^n

Answer 19

r^n(cosnx + isinnx)

Question 20

Volume of Rev with a Parametric Equation

Answer 20

Pi (int (y^2 (dx/dt) dt))

Question 21

Perpendicular Tangent

Answer 21

x = rcosx

Question 22

Parallel Tangent

Answer 22

y = rsinx

Question 23

(z + z^-1)

Answer 23

2cosx

Question 24

(z^n + z^-n)

Answer 24

2cosnx

Question 25

(z - z^-1)

Answer 25

2isinx

Question 26

(z^n - z^-n)

Answer 26

2isinnx

Question 27

How to solve a Partial Fraction with a Quadratic Factor in the denominator?

Answer 27

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)

Question 28

How to find the inverse of a 3x3 matrix?

Answer 28

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.