Pre-Calc Notecards

Examples of how to verify

(tan^2 x + 1)(cos^2 x - 1)=-tan^2 x sec^2 x *-sin^2 x 1/cos^2 x *-sin^2 x -sin ^2 x/cos^2 x = -tan^2 x

Symbol for a vector

---> PQ

1st quadrant

0-90 degrees or 0-π/2 radians

3rd quadrant

181-270 degrees or π-3π/2 radians

4th quadrant

271-360 degrees or 3π/2-2π radians

Example of how to solve equations using identities

2sin^2 x + 3cos x - 3 = 0 2(1-cos^2 x)+ 3cos x - 3 = 0 2 - 2cos^2 x + 3cos x - 3 = 0 2cos^2 x - 3cos x + 1 = 0 (2cos x - 1)(cos x - 1) cos x=1/2 cos x=1 x=π/3,5π/3,0 + 2πn

Angular Speed Formula

2π*# of revolutions/time

2nd quadrant

91-180 degrees or π/2-π radians

Component form of a vector

<x2-x1, y2-y1>

degree

A segment of a circle equal to a 360th of the whole

Graphing

Determine the amplitude Determine asymptotes: Tangent: Set bx + c = - π /2 bx + c = π /2 Cotangent:Set bx + c = 0 bx + c = π Graph: axes, asymptotes, waves (2 full), key points (6 in all)

direction angle

For a Vector that is: v = ll v ll < cos θ , sin θ > = ll v ll ( cos θ ) i + ll v ll ( sin θ ) j the direction angle is: tan θ = sin θ / cos θ = b/a

Conversion of degrees to radians

Multiply degree measure by π/180

Conversion of radians to degrees

Multiply radian measure by 180/π

Sine

Opposite/hypotenuse

linear speed formula

R*2π*# of revolutions/time

SOHCAHTOA

SIN (Opposite/Hypotenuse) COS (Adjacent/Hypotenuse) TAN (Opposite/Adjacent)

Inverse Trigonometric Functions

Sin: y = arcsin x y = sin^-1x Cos: y = arccos x y = cos^-1x Tan: y = arctan x y = tan^-1x

Trig functions of any angle

Sin=y/sqrt(x^2+y^2) Cos=x/sqrt(x^2+y^2) Tan=y/x Csc=sqrt(x^2+y^2)/y Sec=sqrt(x^2+y^2)/x Cot=x/y

Fundamental Trig Identities

Sinx=1/csc x Cosx=1/sec x Tanx=1/cot x Cscx=1/sin x Secx=1/cos x Cotx=1/tan x Tanx=sin x/cos x Cotx=cos x/sin x

Arc length

The length of a portion of the arc of a circle

Radian

The measure of a central angle that intercepts an arc with length equal to the radius of the circle

angular speed

The rate at which an object rotates or revolves; measured in rotations or revolutions per second

dot product

Two vectors are multiplied to produce a scalar not a vector. The dot product of two vectors, u and v, is expressed by the equation u*v = u1v1 + u2v2

Verifying Identities

Work only with one side of the equation (never move anything from side to side) Look for factoring, adding fractions, FOIL, or common Look for ways to substitute If all else fails, convert everything to sines / cosines

Law of Sines

a/sinA = b/sinB = c/sinC AAS,ASA,SSA SSA (ambiguous case), 1,2 or no solutions

Law of Cosines

a^2=b^2+c^2-2(b*c)cosA b^2=a^2+c^2-2(a*c)cosB c^2=a^2+b^2-2(a*b)cosC SSS,SAS

addition of vectors

add components by moving vectors so initial point of one meets the terminal point of the other

Cosine

adjacent/hypotenuse

Cotangent

adjacent/opposite

standard position

an angle with its vertex at the origin and its initial side along the positive x-axis

coterminal angles

angles that have the same initial and terminal sides

Negative angles rotate...

clockwise

Positive angles rotate...

counterclockwise

linear speed

distance traveled per unit of time

Secant

hypotenuse/adjacent

Cosecant

hypotenuse/opposite

Tangent

opposite/adjacent

Arc Length Formula

s=r(theta)

Examples of how to use identities

sec^2*θ - 1 (sec θ - 1)(sec θ + 1) tan^2 θ sin x cos^2 x - sin x sin x(cos^2 x - 1) sin x*-sin^2 x = -sin^3 x

Sum and Difference Formulas

sin (x + y) = sin x cos y + cos x sin y sin (x- y) = sin x cos y - cos x sin y cos (x + y) = cos x cos y - sin x sin y cos (x - y) = cos x cos y + sin x sin y tan (x + y ) = (tan x + tan y)/(1 - tan x tan y) tan (x - y) = (tan x - tan y)/(1 + tan x tan y)

Trig Identities

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

Half-Angle Formulas

sin x/2 = + or - √(1 - cos x)/2 cos x/2 = + or - √1 + cos x)/2 tan x/2 = (1 - cos x)/(sin x) or (sin x)/(1 + cos x)

Double Angle Formulas

sin(2x)=2sin(x)cos(x) cos(2x)=cos^2-sin^2 =1-2sin^2 =2cos^2-1 tan(2x)=2tanx/1-tan^2

Pythagorean Identities

sin^2+cos^2=1 tan^2+1=sec^2 1+cot^2=csc^2

Absolute value of a complex number

the distance from the origin in the complex plane l a + bi l = √ (a^2+ b^2)

Angle between two vectors

u*v = ll u ll ll v ll cos θ cos θ=u/||u||*v/||v||

Trig form of a complex number(Polar Form) and shortcuts for multiplication and division

z = a + b i = (r cos θ ) + ( r sin θ ) z1 z2 = r1 r2 ( cos (θ1 + θ2) + i sin (θ1 + θ2) z1/z2 = r1/r2( cos(θ1 - θ2) + i sin (θ1 - /θ2)

DeMoivre's Theorem

z^n = r^n ( cos(θn)+isin(θn))

unit vector

||v||=1 , v/||v||