OCR MEI Core Pure

∫tan x dx

ln|sec x| + C

∫sec x dx

ln|sec x + tan x| + C

∫sec x tan x dx

sec x + C

∫cosec x dx

ln|cosec x - cot x| + C

∫cosec x cot x dx

-cosec x + C

∫cot x dx

-ln|cosec x| + C

d/dx [sec x]

sec x tan x

d/dx [cosec x]

-cosec x cot x

d/dx [cot x]

-cosec²x

general rotation matrix

8 arrangements for 3 planes in 3d

det M ≠ 0:

• one singular point of intersection

det M = 0:

• 3 parallel (distinct)

• 3 coincident

• 2 parallel, 1 intersects both

• 2 coincident, 1 intersects both

• 2 coincident, 1 parallel (distinct)

• triangular prism

• sheaf

sheaf

3 planes intersecting along a common line (line of infinite solutions)

types of solutions for planes in 3D

unique solution (1 point of intersection)

line of infinite solutions (sheaf)

plane of infinite solutions (3 coincident)

no solutions (all others)

invariant lines vs line of invariant points

line of invariant points - x,y maps to x,y. points off the line will be mirrored

invariant lines - point on the line maps to another point on the same line