math final

Arithmetic Sequence Formula (An)

$$A_n = A_1 + (n - 1)d$$

Sum of Arithmetic Series (Sn)

$$S_n = \frac{n}{2} (A_1 + A_n)$$

Geometric Sequence Formula (An)

$$A_n = A_1 \times r^{(n - 1)}$$

Sum of Geometric Series (Sn)

: $S_n = A_1 \frac{1 - r^n}{1 - r}$ (if $r \neq 1$).

Rose Curve

$r(\theta) = a \cdot \sin(k\theta)$ or $r(\theta) = a \cdot \cos(k\theta)$ where k determines the number of petals.

If k is even 2k petals, if odd k petals

Cardioid/Limacon

$r(\theta)=a(b+\cos(\theta))$ or $r(\theta)=a(b+\sin(\theta)).$

sin anchored on x, cos anchor on y

if a>b no loop, a<b inner loop, a=b dimple

Lemniscate

$r^2 = a^2 \cos(2\theta)$ or $r^2 = a^2 \sin(2\theta).$

sin anchored pie/4, cos anchored x axis

Circle Formula

a(sin x)=r anchored on y

a(cos x)=r anchored on x

Double Angle Identity for Sine

The double angle identity for sine is: $\sin(2x) = 2 \sin(x) \cos(x).$

Double Angle Identity for Cosine

The double angle identity for cosine is: $\cos(2x) = \cos^2(x) - \sin^2(x) = 2\cos^2(x) - 1 = 1 - 2\sin^2(x).$

Sum of Angles Identity for Sine

The sum of angles identity for sine is: $\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b).$

sum of angles for cos

cos(a+b)= cos a • cos b - sin a • sin b