Maths: Trig Identities and sequence expressions and mechanics equations

sin²x+cos²x

1

1+tan²x

sec²x

1+cot²x

cosec²x

sin2x

2sinxcosx

cos2x

cos²x-sin²x or 1-2sin²x or 2cos²x-1

segment area (badge)

½ r²(x-sinx)

AP nth term

a+(n-1)d

GP nth term

ar^n-1

When f^’’(x)<0

Concave

When f^’’(x)>0

Convex

Vectors with constant speed

r=r₀+vt

Vectors with constant acceleration (3 equations)

v=u+at

r=r₀+ ut+ ½ at²

r=r₀+vt- ½ at²

modelling assumptions (5 examples)

• smooth pulley

• string is light and inextensible

• modelled as a particle (consider size/dimensions)

• rod is rigid (won’t bend and has no thickness)

• air resistance

Equation to find the angle a vector makes with the positive x/y axis

cosx=x or y/[a]

When is a binomial valid?

[ax]<1 so [x]<1/a

Sum proof for GP

Sn= a + ar + ar² + ar³+… ar^n-1

rSn=r(ar + ar²+ ar³+… ar^n-1)

rSn= ar + ar² + ar³+… ar^n-1+ arⁿ

Sn-rSn= a-arⁿ

Sn(1-r)= a(1-rⁿ)

Sn=a(1-rⁿ)/1-r

Sum proof for arithmetic

Sn= a, a+d, a+2d+… a+(n-1)d

+Sn= a+(n-1)d, a+(n-2)d+… a+2d, a+d, a

2Sn=n[a+(n-1)d]

Sn=n/2[a+(n-1)d]

Find range when domain is:

Differenciate 2^x

ln2.2^x

Integrate 2^x

(1/ln2).2^x +c

Integrate e^kx

(1/k)e^kx+c