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| Linear and Angular Velocity |
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| sina=y/r cosa=x/r tana=y/x csca=r/y seca=r/x cota=x/y |
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| The Reciprocal Identities |
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| csca=1/sina seca=1/cos cota=1/tana |
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| What are the three sides of a 45degree triangle in standard position |
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| What are the three sides of a 30,60,90degree triangle in standard position |
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| In what quadrants are sine, cosine, and tangent positive and negative? |
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| all in 1, sin in 2, tan in 3, cos in 4. |
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| (ex) what is the linear velocity in mph of the tip of a 22in blade that is rotating 2500rpm? |
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| (ex) what is the linear velocity in mph of a point on the equator? (eg=3950mi) |
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| In a right triangle how can you find the adjacent? the opposite? and hypotenuse? |
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| A=the same side that the angle and the 90degrees are on. O=is the side that is opposite to the angle a. H=the side opposite to the 90degree angle. |
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| (rule) In order to solve a right triangle. . . |
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| Use Pythagorean theorem to find the length of the 3rd side (most of the time r) then use the trig. ratios to find missing angles (a+b+o=180degrees) |
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| (ex) find the height of the top of a tower from point A on the ground is 19.9degrees. from point B is 50ft closer to the tower, the angle of elevation is 21.8degrees. |
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| (ex) a spy plane flies over the soviet union at an altitude of 14mi, how wide a path on the surface of the earth can the spy plane see. (e.r.=3950mi) |
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| In an acute right triangle, sin,cos,tan,csc,sec,andcot are. . . |
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| sina=o/h cosa=a/h tana=o/a csca=h/o seca=h/a and cota=a/o |
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| (ex) sin0 sin30 sin45 sin60 sin90? |
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| 0, 1/2, root2/2, root3/2, 1 |
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| (ex) cos0 cos30 cos45 cos60 cos90? |
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| 1, root3/2, root2/2, 1/2, 0 |
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| What is the fundamental Identity of Trigonometry? |
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| How would youfind the reference angle |
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| this will be the positive acute angle formed on the terinal side. |
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Is this an example of a reference angle, 120degrees equals 60degrees'
235degrees |
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| What is the terminal side? |
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| the side that the angle is going towards, (ex) 35degree it will be the top ray. -35degrees it will be the bottom ray. |
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| (ex) on the unit circle for sine, cosine what is 0degrees, 90degrees, 180degrees, 270degrees. |
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| (1,0) (0,1) (-1,0) (0,-1) |
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| On the unit circle what is 30degrees, 45 degrees, and 60degrees for sin and cos. |
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| (root3/2,1/2) (root2/2,root2/2) (1/2,root3/2) |
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| What are the 5 starting points of the general sine and cosine wave. |
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| x=(0) (pi/2) (pi) (3pi/2) (2pi) |
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| What is the period of a Sine function |
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| |A| (absolute value)in the equation y=Asin[B(x-C)]+D |
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in the equation y=Asin[B(x-C)]+D P.S. is C
(note: C is generally going to be the opposite of what it seems because in the equation it is (-)) |
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| What should you do first in the equation y=2sin(3x+pi)+1? |
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| convert to standard form, y=2sin[3(x+pi/3)]+1 |
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| What is the frequency of a wave? |
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| What is the major differentiation between secant and cosecant graphs from sine and cosine? |
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| These graphs will have asymptotes at pi/2 and 3pi/2. |
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| If given a secant or cosecant equation to graph then you should... |
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| Treat them like cosine and sine graphs then draw in the asymptotes. |
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| What is the asymptote of y=tan(x) |
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| What are the five key points on the tangent graph? |
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| (-pi/2), (-pi/4), (0), (pi/4) (pi/2) |
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| In the graphing procedure of tangent how do you find the three key points? |
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| In the graphing procedure of sine and cosine what is method of finding the 5 key points? |
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| What is the period of a Tangent function |
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| Is there an Amplitude for secant, cosecant, tangent, and cotangent graphs? |
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| No, because they have asymptotes and therefore cannot have a maximum and minimum of the y cordinates. |
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| (ex) A wheel is rotating at 7rads/sec and the wheel has a 100in diameter. what is its linear velocity in the nearest foot. |
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| Reciprocal Identities (8) |
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sinx=1/cscx cosx=1/cosx tanx=sinx/cosx tanx=1/cotx cscx=1/sinx cotx=cosx/sinx
secx=1/cosx |
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| Pythagorean Identities (3) |
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| sin^2x+cos^2x=1 1+cot^2x=csc^2x tan^2x+1=sec^2x |
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| When trying to verify identities you should always work on one side (t/f) |
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| When converting a rational function try multiplying by the numerator (t/f) |
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| If you have a sum or difference try combining them (t/f) |
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| Cosine of a sum and difference (2) |
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cos(a+b)=cosacosb-sinasinb
cos(a-b)=cosacosb+sinasinb |
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| Cofunction Identities (6) |
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sin(pi/2-u)=cosu cos(pi/2-u)=sinu
tan(pi/2-u)=cotu cot(pi/2-u)=tanu
sec(pi/2-u)=cscu csc(pi/2-u)=secu |
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| Sine of a sum or difference (2) |
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sin(a+b)=sinacosb+cosasinb
sin(a-b)=sinacosb-cosasinb |
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| Tangent of sum or difference (2) |
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tan(a+b)=(tana+tanb)/(1-tanatanb)
tan(a-b)=(tana-tanb)/(1+tanatanb) |
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| Double Angle Identities (5) |
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sin2x=2sinxcosx tan2x=2tanx/1-tan^2x
cos2x=cos^2x-sin^2x
cos2x=2cos^2x-1
cos2x=1-2sin^2x |
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sin(x/2)=-+root(1-cosx/2)
cos(x/2)=-+root(1+cosx/2)
tan(x/2)=-+root(1-cosx/1+cosx)
tan(x/2)=sinx/1+cosx
tan(x/2)=1-cosx/sinx |
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| What is tangent of 0, 30, 45, 60, and 90degrees? |
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| 0, root3/3, 1, root3, n.s. |
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| When evaluating inverse functions you must remember that . . . . |
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| they must be inside -pi/2 and pi/2 |
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| What is a composition of a function |
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| an inverse function inside of a function or vice versa. |
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When solving a large trig equation you should. . . .
ex) 2sin2a-root3=0 |
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you should eliminate as much as possible and solitate the angle and the function.
2a=arcsinroot3/2 |
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What is the solution set in a cosx=a equation?
what is it when cosx=1,0,-1 |
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{x|x=s+2kpi}
2kpi
pi/2 + kpi
pi + 2kpi |
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When solving for sinx=a the solution set is. . . .
and when sinx=1,0,and -1? |
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{x|x=s+2kpi}
pi/2 + 2kpi
kpi
3pi/2+2kpi |
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| When solving for tangent equation the solution set is . . . and. . . |
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Definition
{x|x=s+kpi}
{x|x=s+pi+kpi} |
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| When there is multiple angles in a solution then you must. . . |
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| you must divide or multiply the solution sets |
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| What is the ambiguous case? |
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| ssa,there can be 0,1,or2 triangles in the cases. |
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| the direction of the ray. |
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a^2=b^2+c^2-2bccosa
b^2=a^2+c^2-2accosa
c^2=a^2+b^2-2abcosc |
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| What is the length of a chord |
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| a=r(root2-2cosa) take each side and divide it into 360degrees |
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| A=bcsina/2 acsinb/2 absin/c |
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| What is heron's area formula? |
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A=root(S(S-a)(S-b)(s-c))
S=(a+b+c)/2 |
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| To get the sum of the resultant you must follow the . . . . |
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| How do you find the horizontal and vertical componets from vector and direction angle |
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| How to find the magnitude and direction angle from component form |
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| find magnitude by taking the root of this is M now take arcsiny/M then subtract this from 180, this is your angle |
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| How do you find the component form from M and direction |
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| when performing operations with vectors remember. . . |
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| dot product and adding a1withb1and a2withb2 |
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| How do you find the angle between two vectors |
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| when taking <-2,6> into -2i+6j |
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| What is De moivre's theorem |
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| z^n=r^n(cos n a + isin n a) |
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| What is the comples conjugate? |
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| What is the trigonometric form of a complex number? |
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| how do you get theta or (a) in the trig form of a complex number? This is also called the argument. |
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| we use the eguations a=rcosa and b=sina |
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| The first step in De moive's theorem is . . . . |
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| to get it into trig form. |
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