U3 A1 Flashcards
What is the order of pairs in the unit circle (in terms of cos and sin)
(Cos, sin)
What does the radius of the u it circle represent
Hypotenuse of special triangle
What is 0 or 360 degrees on the unit circle
(1,0)
What is 90 degrees on the unit circle
(0,1)
What is 180 degrees on the unit circle
(-1,0)
What is 270 degrees on the unit circle
(0,-1)
Where does the terminal arm always start, and which way does it rotate
Positive x axis
Rotates counterclockwise
What is the angle in standard position
Angle formed between the positive x axis and the terminal arm
What is the “angle in standard position” also called
Rotation angle
What is the reference angle (and how is it written)
Angle between the terminal arm and the closest x axis
Theta sub r
What is always the domain of a reference angle
Greater than on equal to 0 degrees, less than or equal to 90 degrees
How are the quadrants labelled on a graph
Q1 is top right (and label counterclockwise)
Q4 is bottom right
Positive angle: when the terminal arm is rotated in a ______ direction
Counter clockwise
Negative angle: when the terminal arm is rotated in a ______ direction
Clockwise
Coterminal angle: angle that are generated so that the ______arm is in the exact _____ position
Terminal
Same
What is the principle angle
Small positive coterminal angle
What is the formula for conterminal angles (and what do the variables represent)
P + 360 degrees (n), n € I
P=principle angle
n= # of rotations
How do you determine coterminal angles
Add/subtract multiples of 360 from original angle
What is the formula that relates radian to arc length
And what is theta measured in
Theta = a/r
Theta is measured in rads
What is 360 degrees in rads
2(Pi)
What is 180 degrees in rads
Pi
Conversion factor for degrees to rads
1 degree = (Pi/180) rads
Conversion factor for rads to degrees
1 rad = (180/Pi) degrees
What is 30 degrees in rads
Pi/6
What is 45 degrees in rads
Pi/4
What is 60 degrees in rads
Pi/3
What is 90 degrees in rads
Pi/2
WHAT DO YOU HAVE TO MAKE SURE OF WHEN YOUR’RE USING CALC
That it’s in the correct mode
Never put a ____ ratio when taking the inverse to find the angle
(And What do you do instead)
Negative
Use the positive ratio
What’s the ratio for sin theta
y/r
What’s the ratio for cos theta
x/r
What’s the ratio for tan theta
y/x
What is the ratio for csc theta
And what is it the reciprocal of
r/y
Reciprocal of sin theta
What is the ratio for sec theta
And what is it the reciprocal of
r/x
Reciprocal of cos theta
What is the ratio for cot theta
And what is it the reciprocal of
x/y
Reciprocal of tan theta
How do CAST laws apply to reciprocal ratios
follow the same cast rule as their reciprocal ratio
What is (root 3)(root 3)
3
How to solve this:
csc (-5Pi/3)
(And what’s the answer)
1) recognize Pi/3 = 60 degrees
2) -5(60) = -300 degrees
3) draw -300 degrees on graph
4) find reference angle (60 degrees)
5) use hand rules to determine the sin ratio the flip it (since it’s csc)
6) rationalize the denominator and the answer is 2(root3)/3
What is cos theta in terms of the unit circle
X
What is sin theta in terms of the unit circle
Y
What is tan theta in terms of cos and sin
sin/cos
How to label the CAST law
C is in the bottom right quadrant
And start labelling counterclockwise
Steps for finding a trig ratio
1) determine quadrant
2) determine reference
3) determine ratio (finger rules)
4) determine if negative or positive based off CAST law
Finger rule for sin theta
root bottom fingers/2
Finger rule for cos theta
root top fingers/2
What is the finger rule for tan theta
Root bottom fingers/ root top fingers
When asking for sin, cos, tan, theta or reciprocals what are you trying to find
The ratio
All points on the unit circle must satisfy what equation
x^2 + y^2 = 1
What does it mean to solve for
sin theta =1/2
Means find the angle that makes this statement true
Steps for finding angles from a ratio
1) ask yourself in which quadrants is this ratio positive/negative and draw the terminal arm
2) find the reference angle (by taking the inverse of the positive ratio)
3) determine the rotation angles based off the quadrant and the domain
Can there be multiple answers when finding angles from a ratio
(And why)
Yes
Because that ratio can be positive or negative in multiple quadrants, so there are multiple angles
When finding angles from a ratio in rads what do you do differently
After you take the inverse of the ratio divide out pi, then hit math frac
Express answer as that fraction multiplied by pi
How do you solve cos theta = 0
Cos theta = x
So think where on the unit circle does x equal 0
How do you solve
cot theta = undefined
cot is the inverse of tan so whenever tan is 0 (since tan is in the denominator)
What is a periodic function
Graph repress regularly over time
What is amplitude
and is it a vertical or horizontal characteristic
Half way distance between the max and min points, from middle of graph to top
(Vertical)
What is period
And is it a vertical or horizontal characteristic
The interval for one cycle to occur, interval before the graph repeats
(Horizontal characteristic)
The graph of y=sinx
What is the amplitude, period, domain, range, x intercepts, and y intercepts
Amp: 1 unit
Period: 360 degrees or 2Pi
Domain: (-infinite, infinite)
Range: [-1,1]
X intercepts: 180 degrees (n), n € I (infinite)
Y intercepts: (0,0)
Where does a sin graph start
And what does one cycle look like
(0,0)
Starts at zero, zero abs goes up then down then back to x=0
The graph of y=cosx
What is the amplitude, period, domain, range, x intercepts, and y intercepts
Amp: 1 unit
Period: 360 degrees or 2Pi
Domain: (-infinite, infinite)
Range: [-1,1]
X intercepts: 90 degrees + 180 degrees (n),
n€ I (So has to start at 90 degrees)
Y intercepts: (0,1)
Where does the graph of cos start
Abs what does one cycle look like
(0,1)
Starts at the top, goes down past the midline, and comes back up to the top
Where are the vertical asymptotes on y=tanx
And why
Wherever x=0,
so 90 degrees and 270 degrees, negative and positive
(Since tan=y/x undefined values are whenever x=0)
The graph of y=tanx
What is the amplitude, period, domain, range, x intercepts, and y intercepts
Amp: none
Period: 180 degrees
Domain:
Range: (-infinite, infinite)
X intercepts: 180 degrees (n), n € I (infinite)
Y intercepts: (0,0)
The graphs of y=sinx and y=cosx
are identical except for a _____ _____
(And what is it)
Horizontal translation
cosx = sin (x + 90 degrees)
Sinx = cos (x - 90 degrees)
What is the only transformation where x intercepts are invariant
Vertical stretch
Describe the transformation here
y=sin(x - 45 degrees)
Horizontal translation 45 DEGREES to the right (not units)
what transformation would affect amp
Vertical stretch
What transformation would affect period
Horizontal stretch
What transformations would affect range
Vertical stretch and vertical translation
What transformations would affect domain
None
Does reflecting the graph about either axis change any of the characteristics
No
What is the equation of the vertical asymptotes on the graph
y=tanx
x = 90 degrees + 180 degrees (n),
n€ I
What is the middle of a tan graph
The bend
What is the starting point of a tan graph
(0,0)
Equation for period of graph in the form y=a(sin)bx or y=a(sin)bx
2Pi/b
Equation to find period of the graph y=a(tan)bx
Pi/b
How to determine the scale of the horizontal axis
1) determine the distance from 0 to the first know value
2) divide that by the number of increments between
What is vertical displacement also called
Vertical translation
Equation for finding vertical displacement
d= (max+ min)/2
How to find the vertical displacement on a tan graph
Middle of the graph is the bend at the x axis,
so count how many units it is from the original bend to the new bend
What is horizontal phase shift also called
Horizontal translation
What translation is amplitude associated with
Vertical stretch
What is the domain of a transformed graph
(-infinite, infinite)
What is the range of a transformed graph
[ d-a, d+a]
How to find the max and min points on a transformed graph
Max = d + a
Min = d + a
WHAT DO YOU STILL HAVE TO REMEMBER TO DO WITH THE “B” VALUE IN TRANSFORMATION EQUATIONS
Factor it out if needed
IF GRAPHING A FUNCTION WITH A PI CHARACTER WHAT DO YOU HAVE TO DO
Calculator MUST be in radian mode
What is the order when determining an equation from a graph
1) vertical displacement
2) amplitude
3) period, then b value
4) horizontal phase shift
What is the formula to find amplitude on a graph
a = (max-min)/2
How to find period of a graph
1) mark off one cycle
2) count the number of increments between the start of the cycle to the end
3) period= (# of increments)(scale)
If asked for the cos function when you already have the sin function, given a graph, what value(s) would CHANGE
Only “c”
Which calc mode are all application questions done in
Radian
How do you use you calculator and intersection points with trig application questions
1) if given dependent variable (one before equals sign) you can graph
2) y1= # before equals sign
y2= equation after equals sign
3) look at intersection points
On a graph which axis is the dependent variable and which is the independent
Independent- x axis
Dependent- y axis
What is the independent variable in the equation
The one in brackets before the equals sign
Can a sinusoidal function be a tan function
No
Steps for circular motion problems (first three steps)
1) determine min, mid, and max values with simple addition/subtraction and reasoning
2) determine period (stated in question)
3) draw graph
- only 3 values on y scale are min, mid, and max
- only three values on x scale are half the period, the period, abs double the period
- draw at least 9 points on the graph
- sin graph: start at min, max point should be between a period
- cos graph: start at max, min point should be between a period
Drawing a sin graph: start at ___, and _____ should be half (in the middle of), a period
___ point should be at the period
Min
Max
Min
Drawing a cos graph: start at ___, and _____ should be half (in the middle of), a period
___ point should be at the period
Max
Min
Max
Steps for circular motion problems (last three steps)
4) determine a, b, c, d values from graph
- a is the radius but you can still use the formula to determine
5) write a function to model the situation
6) answer what question asks
What variable represents horizontal phase shift
c
What is the form of a transformed sin function
y= a(sin)[b(x-c)] + d
How to determine the c value of a cos graph when a<0
c value is the first min point
How to determine the c value of a cos graph when a>0
c value is the first max point
How to determine the c value of a sin graph when a<0
c value is the first mid point on the DECLINE
How to determine the c value of a sin graph when a>0
c value is the first mid point on the INCLINE
If a reflection about the x axis has occurred it will change your ____ and ___ values
a
c
What is vertical displacement
Middle of the graph
