Trigonometry, Polar Coordinates, and Series Review

Pythagorean Identity (Sine and Cosine)

$$\sin^2(x) + \cos^2(x) = 1$$

Pythagorean Identity (Cotangent and Cosecant)

$$1 + \cot^2(x) = \csc^2(x)$$

Pythagorean Identity (Tangent and Secant)

$$\tan^2(x) + 1 = \sec^2(x)$$

Cosine Sum Identity

$\cos(a+b) = \cos(a)\cos(b) - \sin(a)\sin(b)$ (Flips the sign)

Tangent Sum Identity

$$\tan(a+b) = \frac{\tan(a) + \tan(b)}{1 - \tan(a) \tan(b)}$$

Double Angle Identity for Sine

$$\sin(2x) = 2\sin(x)\cos(x)$$

Double Angle Identity for Cosine (Basic)

$$\cos(2x) = \cos^2(x) - \sin^2(x)$$

Rose Curve Petal Rule (Even n)

If $n$ is even, the graph has $2n$ petals.

Rose Curve Petal Rule (Odd n)

If $n$ is odd, the graph has $n$ petals.

Polar Distance Formula

$$D = \sqrt{r_1^2 + r_2^2 - 2r_1r_2\cos(\theta_1 - \theta_2)}$$

Polar to Rectangular Conversion (x and y)

$x = r\cos(\theta)$ and $y = r\sin(\theta)$

De Moivre's Theorem (Complex Exponent)

$$z^n = r^n(\cos(n\theta) + i\sin(n\theta))$$

Arithmetic Sequence (nth term)

$$a_n = a_1 + (n-1)d$$

Arithmetic Series (Sum formula)

$$S_n = \frac{n}{2}(a_1 + a_n)$$

Geometric Series (Finite Sum formula)

$$S_n = \frac{a_1(1-r^n)}{1-r}$$