AP Precalculus - 3.12 - 3.15 Trigonometric Identities, Complex Numbers, Polar Numbers, and Polar Coordinates (2025)

Pythagorean Identities

sin²θ + cos²θ = 1

1 + tan²θ = sec²θ

1 + cot²θ = csc²θ

sin(90°- θ)

cos(θ)

cos(90°- θ)

sin(θ)

tan(90°- θ)

cot(θ)

sin(-θ)

sin(θ)

cos(-θ)

cos(θ)

tan(-θ)

-tan(θ)

cos(a-b)

cos(a)sin(b)+cos(b)sin(a)

cos(a+b)

cos(a)sin(b)-cos(b)sin(a)

sin(a-b)

sin(a)cos(b)-sin(b)cos(a)

sin(a+b)

sin(a)cos(b)+sin(b)cos(a)

tan(a+b)

[tan(a)+tan(b)]/[1-tan(a)*tan(b)]

tan(a-b)

[tan(a)-tan(b)]/[1-tan(a)tan(b)]

sin(2a)

2sin(a)cos(a)

(Similar to sin(a+b), alternative to remember)

cos(2a)

cos^2(a)-sin^2(a)

1-2sin^2(a)

2cos^2(a)-1

tan^2(a)

[2tan(a)]/[1-tan^2(a)]

(Similar to tan(a+b), alternative to remember)

Laws of Sine

sin(A)/a = sin(B)/b = sin(C)/c

(For Laws of Sine and Cosine, "a, b, and c" mean the side lengths and "A, B, and C" mean the angles correspondent to the side lengths.)

Laws of Cosine

a^2 = b^2 + c^2 - 2cos(A)

b^2 = a^2 + c^2 - 2cos(B)

c^2 = a^2 + b^2 - 2cos(C)

(For Laws of Sine and Cosine, "a, b, and c" mean the side lengths and "A, B, and C" mean the angles correspondent to the side lengths.)

Product of Two Complex Numbers

z1z2 = (r1r2)(cos(θ1+θ2) + i*sin(θ1+θ2))

(z is a complex number)

Quotient of Two Complex Numbers

z1/z2 = (r1/r2)(cos(θ1-θ2) + i*sin(θ1-θ2))

(z is a complex number)

De Moivre's Theorem

z^n = [(r^n)(cos(nθ) + isin(n*θ)]

(z is a complex number)

The n-th root of a complex number

(n root of r)(cos((x+2(pi)k)/n) + i*sin((x+2(pi)k)/n)

k is any integer

Rectangular and Polar Form

Rectangular (x,y)

Polar (r, θ)

Rectangular to Polar Conversions

(x,y) = (rcos(θ), rsin(θ)

r = root(x^2 + y^2)

θ = arctan(y/x) = arcsin(y/r) = arccos(x/r)

Polar Form of Complex Number

z = (rcos(θ) + ri*sin(θ))

Form of Complex Numbers

z = a + bi

graphed in the complex plane a,b for x,y

(a,b are real numbers)

Average Rate of Change of Radius per Radian

[f(θ2) - f(θ1)] / [(θ2) - (θ1)]