exam 3 Flashcards
basic unit circle values
x^2+y^2=1
π/6 (√3/2, 1/2)
π/4 (√2/2, √2/2)
π/3 (1/2, √3/2)
log properties
loga1= 0 logaA= 1 logaA^x=x a^logaX= x logaA^c= ClogaA loga(AB) = logA+logB loga(A/B)= logA-logB
exponential and log equations steps
1) put exponential on one side
2) solve
trig functions
sin^2(x) + cos^2(x)= 1
tan^2(x)+1=sec^2(x)
1+cot^2(x)=csc^2(x)
degrees to radians
multiply π/180
radians to degrees
multiply 180/π
coterminal angles
add or subtract 2π (360 degrees) to given angle
trig functions of angles
SOH CAH TOA
a^2+b^2=c^2
basic trig function domain and range
sin(θ)
domain: (-∞,∞)
range: [-1,1]
cos(θ)
domain: (-∞,∞)
range: [-1,1]
tan(θ)
domain: (-∞,∞) {π/2+nπ}
range: (-∞,∞)
inverse trig functions
sin-1(θ)
domain: [-1,1]
range: [-π/2, π/2]
cos-1(θ)
domain: [-1,1]
range: [0, π]
tan-1(θ)
domain: (-∞,∞)
range: (-π/2, π/2)
cancellation property
f(f^-1(x))= x
f^-1(f(x))=x
*anything >1 is undefined
even-odd identities
sin(-x)= -sinx cos(-x)= cosx tan(-x)= -tanx
guidelines for proving trig identities
1) start with one side
* indicate which side u choose to start with
2) use known identities
3) covert to sines and cosines
formula for sine
sin(x+y)= sinXcosY + cosXsinY
sin (x-y)= sinXcosY - cosXsinY
formula for cosine
cos(x+y)= cosXcosY - sinXsinY cos(x-y)= cosXcosY - sinXsinY
double-angle formulas
sine: sin2x= 2sinxcosx
cosine: cos2x= cos^2x-sin^2x
= 1-2sin^2x
= 2cos^2x-1
tangent: tan2x=sin2x/cos2x
sine graph
sink(x-b)+c period: 2π/k amp: IaI phase shift: b d=period/4 *always starts with "b" *basic graph starts from 0
cosine graph
acosk(x-b)+c period: 2π/k amp: IaI phase shift: b d=period/4 *always starts with "b" *basic graph starts at highest point
tangent graph
atank(x-b)+c period: π/k d=period/4 *b in the middle *2 asymptotes at the very ends
basic trig equation steps
1) find primary solution in one complete period
sin [0,2π) cos [0,2π) tan (-π/2, π/2)
2) find general solution by adding the solution in step 1 by the multiple of the period
*sin and cos: add 2kπ
*tan: add kπ
5-step strategy
1) write down in one function of one angle
2) find values of written function
3) solve for angle
4) solve for variable
5) check restrictions
basic trig equations CHECK
1) factor
2) identities
3) formulas
- addition/subtraction
- double angle
* u substitution
if inverse function is on the outside, look at the
domain
if inverse function is on the inside, look at the
range
solving exponential/log equations: exponential
1) isolate exp
2) take loga
3) solve for variable
* no check
* sometimes we can factor
solving exponential/log equations: log
way 1) 1. isolate loga 2. write in exp form 3. solve for variable 4. check way 2) logaX=logaY; X=Y
