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function
- special kind of relationship
- ratio dependent on angle
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amplitude
height of wave (can't be negative)
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y=a sin x, amplitude
- a
- coefficient of trig function
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period
how long it takes to complete a full cycle
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y=sin bx, period
- b
- coefficient of angle portion
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negative function
flip graph
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steps to graphing function graph
- 1.horizontal translation
- 2. end of full cycle
- 3. half of cycle
- 4. quarter of cycle
- 5. all other tickmarks
- 6. new x-axis (v.t.)
- 7. max value
- 8. min value
- 9. graph & check
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steps to graph csc & sec
- 1. use sin/cos as guide function
- 2. sketch sin/cos graph
- 3. sketch vertical asymptotes
- 4. sketch csc/sec graph
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steps to graph tan
- 1. determine period
- 2. sketch vertical asymptotes
- 3. sub-divide the interval into four equal parts
- 4. find values at 1st qtr, midpoint, 3rd qtr point
- 5. join points with smooth curve that approaches vertical asymptotes
- 6. check values
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steps to graph cot
- 1. determine period
- 2. sketch vertical asymptotes
- 3. sub-divide the interval into four parts
- 4. find values at 1st qtr point, midpoint, 3rd qtr point
- 5. join points with smooth curve
- 6. check values
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reciprocal identities
- sin theta= 1/ csc theta
- csc theta=1/sin theta
- cos theta=1/sec theta
- sec theta=1/cos theta
- tan theta=1/cot theta
- cot theta=1/tan theta
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quotient identities
- tan theta=sin theta/cos theta
- cot theta=cos theta/sin theta
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pythagorean identitites
- sin2theta+cos2theta=1
- 1+tan2theta=sec2theta
- cot2theta+1=csc2theta
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negative angle identities
- sin theta=-sin(-theta)
- cos theta=cos (-theta)
- tan theta= -tan (-theta)
- csc theta= -csc (-theta)
- sec theta= sec (-theta)
- cot theta= -cot (-theta)
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hints for verifying identities
- 1. make more complicated (longer) side simpler (shorter)
- 2. change all trig functions to sine and cosine, then simplify
- 3. factor
- 4. side you're not changing is your goal
- 5. if expression has 1+sin x, multiplying numerator & denominator by 1-sin x would give cos2x
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difference identity for cosine
cos(A-B)=cosA cosB + sinA sinB
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sum identity for cosine
cos(A+B)=cosA cosB - sinA sinB
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cofunction identities
- cos(90-theta)=sin theta
- sin(90-theta)=cos theta
- tan(90-theta)=cot theta
- cot(90-theta)=tan theta
- sec(90-theta)=csc theta
- csc(90-theta)=sec theta
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sum identitiy for sine
sin(A+B)= sinA cosB + cosA sinB
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difference identity for sine
sin(A-B)= sinA cosB - cosA sinB
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sum identity for tangent
tan(A+B)= tanA + tanB / 1- tanA tanB
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difference identity for tangent
tan(A-B)= tanA - tanB/ 1+ tanA tanB
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double angle identities for cosine
- cos2A= cos2A - sin2A
- cos2A= 2cos2A - 1
- cos2A= 1 - 2sin2A
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double angle identity for sine
sin2A= 2sinAcosA
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double angle identity for tangent
tan2A= 2tanA / 1-tan2A
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half angle identity for cosine
cos theta/2= +/- root 1+cos theta/2
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half angle identity for sine
sin theta/2= +/- root 1-cos theta/2
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half angle identities for tangent
- tan theta/2= +/- root 1-cos theta/1+cos theta
- sin theta/ 1+cos theta
- 1-cos theta/ sin theta
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