Trig Equations for Final Exam

sin =

opp/hyp

cos =

adj/hyp

tan =

opp/adj

csc =

hyp/opp

sec =

hyp/adj

cot =

adj/opp

quotient identities : tan =

sin/cos

quotient identities : cot =

cos/sin

Pythagorean Identities

sin² + cos² = 1 / 1+ tan² = sec² / 1 + cot² = csc²

Reciprocal Identities

sin = 1/csc, cos = 1/sec, tan = 1/cot

Cofunction Identities

sin(90° - x) = cos(x), cos(90° - x) = sin(x), tan(90° - x) = cot(x)

Distance Formula

r = √((x² - y²)

Trigonometric Functions for General Angles

sin = y/r , cos = x/r , tan = y/x , csc = r/y, sec = r/x, cot = x/y

Area of a Triangle

A = 1/2absin(C)

Reference Angles

Q1 = θ, Q2 = pi - θ, Q3 = θ - pi, Q4 = 2pi - θ

Polar Coordinates

r = √((x² - y²) , θ = tanθ_r= |y/x|

Law of Sines

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

Law of Cosines

a² = b² + c² - 2bc cos(A) , a²+b²- c / 2ab = cos(c)

rcosθ = a

Vertical Line

rsinθ =a

Horizontal Line

ar cosθ + br sinθ = c

polar coordinates representing a line at an angle.

θ = a

line through pole

r = a

represents a circle with radius a centered at the pole.

r = a cosθ

represents a circle with radius a centered at the polar axis.

r = a sinθ

represents a circle with radius a centered at the polar axis, shifted vertically.

r = a +bsin(θ) and a + bcos(θ)

cardioid or limacon

|a/b| = 1

cardioid

|a/b| < 1

limacon with an inner loop

1 < |a/b| <2

limacon with a dimple

|a/b| >= 2

limacon with no dimple or inner loop

r = a sin(nθ) r = a cos(nθ)

Roses

r = a sin(nθ) where n is odd

rose has n petals and endpoint lays along vertical line θ = pi/2

r = a cos (nθ) where n is odd

rose has n petals and endpoint lies on either polar axis or line θ = pi/2

r = asin(nθ) where n is even

petals are 2n and none of the endpoints land on polar axis or line pi/2

r = acos(nθ) where n is even

rose has 2n petals and endpoints have two petals between the polar axis and the line θ = pi/2

r² = a² sin (2θ) and r² = a² cos (2θ)

lemniscate

r² = a² sin (2θ)

lemniscate symmetric about the pole and the line θ = pi/4 and the endpoints of the two loops occur when θ = pi/4 and θ = 5pi/4 and the lengths of the loops is |a|

r² = a² cos (2θ)

lemniscate symmetric about the pole, the horizontal line θ = 0, and the vertical line θ = pi/2, the endpoints of the two loops occur when θ = 0 and θ = pi, the length of the loops is |a|

heron’s formula

A = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter.