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||