• Trig Notes--Fall Semester
     
    Trig1ChapterP "Prerequisites"
    Trig2A  "Trig Intro."
     

     
    Trigonometric IDs

     
     boromir
     
    Trig Formulas  
     
    joker literal EQ
     
     
     ANGLE SUM AND DIFFERENCE FORMULAS
    Chapter 4 Lesson 4
     
     
    Since 15 degrees are missing angles from the unit circle,
    let's try calculating sin 15 and cos 15.
     sin15 and cos15
     
     
    Use this rectangle to derive sin75 and cos75.
    cos75 and sin75  
     
    Using a different diagram allows derivation of general formulas
    for the sum and differences of angles.
    Addition Formulas Diagram
    All we have to do is find the missing sides in terms of sines and cosines.
     
    Generalizing the diagram enables derivation of the formulas:
    Angle Sum Diagram
    From cosAcosB = cos(A+B) + sinA sinB we have:
    SUM:   cos (A+B) = cosA cosB - sinA sinB
    SUM:   sin (A+B) = sinA cosB + cosA sinB
     
     
     
    Angle Difference Diagram
    From adding and subtracting segments to congruent segments, we have:
    COSINE DIFFERENCE:  
    cos(A-B) = cosA cosB + sinA sinB
     
    SINE DIFFERENCE:  
    sin(A-B) = sinA cosB - cosA sinB
     
     
     
    sum and difference diagrams  
     
    You can also derive the sine angle addition formula from Area = 1/2 bc sinA of a triangle.
    sine addition triangle  
    1/2 dc sin(A+B) = (d*sinB) (c*cosA) /2 + (c*sinA) (d*cosB) /2 giving
    sin(A+B) =  sinB cosA + sinA cosB, which can be re-arranged to become:
     
    sin(A+B) =  sinA cosB + cosA sinB
     
     
     
    PYTHAGOREAN IDENTITIES
     Chapter 4 Lesson 1
     
    Let a = sinθ, b = cosθ and c = 1.  Then by the Pythagorean Theorem,
     
    sine squared plus cos squared  
    Note the sine and cosine segments are perpendicular.
     
    We can obtain two other pythagorean identities by dividing each addend:
    over sine squared   to get
     
     over cos squared
    Here's how the related triangles appear on the unit circle:
     
     three pythagorean identity circles
    Notice the tangent touches the circle once and the secant will cut it twice.
     
    Placing all the triangles on the same unit circle yields:
     six trig segments
    Note the tangent and cotangent segments are perpendicular.
     
    We can also rotate the triangles so that the right angle is tangent to the unit radius.
    tangent segment identities
     This time the cosecant and secant segments are perpendicular.
     
     Find the segments lengths of theta θ is 45 degrees.
     trig circle segments
     
     Find the coordinates below if theta θ is 60 degrees.
    trig segment coordinates  
    P(   ?,   ?);    R(   ?,   ?);      Q(   ?,   ?)
     
     The other segments formed have names as well.
    trig circle segments
     
    Here's a way to remember the identities:
    magic hexagon
     
     
     
     
    SOLVING TRIANGLES
     
    Use Law of Cosines for SSS cases when given sides a, b, and c;
    and SAS when given a, ∠B, and c.
     
    Use Law of Sines for ASA (given ∠A, c, and ∠B with ∠C = 180 -∠A-∠B)
    and AAS (A, ∠B, and a) cases.
     
    When solving triangles with the SSA case of Law of Sines,
    one of three things can happen.
    This ambiguity is therefore called "The Ambiguous Case."
     
     ambiguous cases
     
    ambigous supplements
     
    ambiguous 2 cases
    This is why there is  no Angle Side Side triangle congruence--two triangles.
     
     ambigous one
    Note: in this case SSA becomes HL (Hypotenuse-Leg congruence.)
     
    ambigous no solution
     
     
    ambigous none  
    For a more detailed explanation with numbers, check this link out:
     
     ambigous no triangle
     
    In summary when doing a problem:
    If sine < 1 there are two solutions
    (or one if the angle you get is less than the given angle. You can check by seeing if the supplement violates the triangle sum of 180).
    If sine = 1, there is one solution.
    There are no solutions if sine > 1.
     
     ambiguous cases
    (a) sine > 1               (b) sine = 1               (c) sine < 1, a < b               (d) a > b, S.s.A. case => 1 triangle
     
    high pot in use  
     

     
    Star Trig   math frontier
     
    planet alert
     
    planet
     
    planet help
     
    instructions
     
    planet height
     
    planet distance
     
    planet ship
     
    planet solution
     
    transporter Energize
     
     
    Trigonometry Theorems  
     
    tan(gerine)
     
    Trig Theorems  
     
    Wolowitz Theorem
     
     
     Unit 2B Lesson 1.7
        Arcsine and ArcCos
    arcsin worksheet
     
    seven sins  
     
    sine theta arcsin y
     
     
    arccosine worksheet  
     
    cos theta arccos(y)  
     
    arccosine angles   arccos dom range
     
    So for the range of the inverse trig functions we have:
    inverse trig ranges  
     
    Now let's graph the data points for these inverse trig functions.
     
    arcsine angles Graphing these coordinates (t, y) and (y, t) yields:
     
    arcsin sin graphs
     
    Graphing (t, x) and (x, t) gives us the arccos graph.
     
    arcsin sine graphs
     
     
     
    arcsin graph   arccos graph
     
    arccos cosine graphs  
     
    graphs sin cos arcsin arccos
     
    arcsine graph   arccosine graph
     
    The graphs of all six inverse trig functions follow:
    trig inverse graphs  
    NOTE:  The inverse cotangent pictured is wrong.
     

     
    Unit 2B Lesson1.5
     
    sine graph
    sine on 0 to pi sine on pi to 2pi
     
    sign graph  
     
    stop sine  sine muffins
     
    sine biscuits
     
     
     
    Unit 3 Notes 3 (identities)
     
     
     
    DRG Key  
     
    rad time  
     
    Degrees Minutes Seconds
    6.a) 3600 seconds   b) 1/60 = 0.01666...   c) 1/3600 = 0.0002777...  
    d) 5 + 0.5 + 0.00166... = 5.501666...
    e)  89 + 0.3 + 0.011666... = 89.3116666...
     
    7.   a) 57 degrees + (0.29578)60 minutes = 57 deg 17.746 minutes = 57 deg 17' 60(0.746) seconds
       = 57 deg 17' 45" (rounding 44.8 seconds)
     
    8. a) 400 grad / 360 deg = 10/9 = 1 1/9 = 1.111...   b) 90 deg / 100 grad = 9/10 = 0.9
        c)  30 * (10/9) = 100 / 3 = 33 1/3 = 33.333... gradians
        d)  66 2/3 * 90 deg / 100 grad = (200/3)*(9/10) = 180/30 = 60 degrees
        e) 1 grad * 9/10 = 0.9 degrees = = (0.9) * 60' = 54 minutes
        f )  25 grad * 54 min = 1350 min / 60 deg = 22.5 degrees = 22 degrees 30 seconds
       or  25*9/10 = 22.5
       
    Pythagorean IDs
    Pythag ID 4
     
     Pythag ID 5
     
    Domain Ranges  

    Let's define sine and cosine functions from their unit circle ratios.
    Using the angle measures in degrees,  observe the Quadrants:
    Quadrant I: 0 < θ < 90, Quadrant II: 90 < θ < 180,  Quadrant III: 180 < θ < 270,  Quadrant IV: 270 < θ < 360.
     
    sine and cosine
    Note the Quadrants where both sine and cosine are positive (+, +), negative (-,-) or opposites (-,+) & (+,-).
    Also observe how sine starts at 0 and cycles through all the values ending in 0 again after 360 degrees.
    This is the period of the sine function.  In radians, the period is 2π.
    Likewise the cosine starts at 1 and goes down to -1 before ending at 1 again after 2π radians. Its period is also 360°.
     
    Since the angle can rotate around the circle infinitely many times,
    both in positive counter-clockwise directions and negative clockwise directions,
    the domain of each is the set of all Real numbers:  -∞ < t < ∞ or (-∞, ∞).
     
    As all the fractions are between 1 and -1, the range of each function [-1 , 1].
    For y(t) = sin(t) the range is -1 ≤ y ≤ 1 and for x(t) = cos(t) the range is -1 ≤ x ≤ 1.
     
    Since they are right triangles, apply the Pythagorean Theorem.
    Pythagorean IDs
     So there are three Pythagorean identities.
     
     
    Explore the relationship of the new functions:  secant, cosecant, and cotangent.
    Reciprocal Cofunction IDs  
    Not only are the cofunctions reciprocals of sine, secant, and tangent,
    they are related via complementary angles.  Hence, the prefix "co":
    cosine(angle) = sine(complement),
    cotangent(angle) = tangent(complement) and
    cosecant(angle) = secant(complement.)
     
    Likewise sin(angle) = cos(complement), tan(angle) = cot(complement) and sec(angle)= csc(complement).
     
     Dude rationalize the denominator
     
    Unit 2 Notes 3 (six trig ratios)
     trig dice
      syrcxrtyx ratios in Qii
    3-4-5 triangle 3-4-5 ratios
     
    secant ratios in Qi  secant UC ratios
     
     cosign
     
    special triangles  special unit triangles
     
     
    height 30 height 45 height 60
     
    sin 45  sin 60
     
    cos 45   cos 60
     
     
    Unit 2 Notes 2 (Unit Circle coordinates)
     
     
    Since dividing the unit circle into eight and twelve equal sectors results in angles that are multiples of 30, 45, and 60 degrees, the side of special right triangles provide the (x, y) coordinates on the unit circle.
     
    Recall the 30-60-90 triangle and the 45-45-90 triangle.
     
     special unit triangles
     
    We divide each side by the length of the hypotenuse so that the newer triangles can fit into the unit circle of radius 1.
    So the 30-60-90 triangle gets divided by 2, and the 45-45-90 triangles gets divided by √2.
     
    Now let's fit a 30-60-90 triangle into the first Quadrant of the unit circle.
     
     triangle 60
     
     Then fit each new special triangle all around the unit circle.
     unit circle blank
    By accounting for positive (+) and negative (-) signs in each quadrant, we can find the (x, y) coordinates all the way around the circle.
     
    triangles across axes  
     
    Filling in the coordinates all the way around the circle we now have:
     Unit Circle in color
     
    Unit 2A Chapter 1.1-1.4
    Lesson 1.1
    Notes #1 Radian Measure
    Eight Slices  twelve slices
     Notes #2 9/18/2012
    Radian Measure  
     
    Standard Position  
     
    Negative angle measures  
     
    DMS Example 1  DMS Example2
     
     
    Unit 1 Chapter P
    "Prerequisites"
     
     Click on the link below for each lesson's notes:
     
     
     
     
    August 21, 2013
    A scene in the movie Smilla's Sense of Snow compares human life to the number system:
     
    NZQRC  Number Sets
    Sets Open Closed  absolute value def
     
    August 22, 2013
    Solving a linear equation:
    0.99999...=1  
     Factoring a quadratic with a leading coefficient:
    Diamond Box factor  
    Solving a rational equation:
    Rational EQ  
    Check for extraneous solutions.  -2 makes the expression undefined (dividing by 0).
     
    Radical Equation (square both sides to eliminate the root)
    Radical EQ  
    Once again, check the solutions.
    If there are two radicals, square both sides twice!
    Two Radicals  
    and check the solutions.
     
     August 26, 2013
    "Cartesian Plane"
     
    Cartesian Plane  
     
    Circle Equation  
     
     Circle Example
     
    Linear Equations
     point slope form
    table example
    Example 4
    Example 5
    II Practice  
     
    Notes #5 -- "Functions"
    August 29, 2012
     The definition of a function is written in blue.
     Function Definition
    A function can map numbers from one set to another via a diagram,
    Celsius Fahrenheit sets  
    or a table, or it can relate input and output values through an equation, for example y = |x|.
    absolute value  Domain and Range
    You can tell the graph is of a function because it passes the vertical line test.
    You can tell from the table by checking if there are repeat values for x.  In the case above, there are not.
    In the case below, there are repeat values of x.
     circle per3  circle per2
    This circle is not a function because some x-values are associated with two y-values.
     So the graph fails the vertical line test and there are repeat values of x in the table.
    The domain and range are the same in this relation: {0, 3, 4, 5, -4, -3, -5}.
    These are discrete points since we can count them.  Find the circle's equation to connect its dots.
    circle D and R  circle equation
    For the equation of the circle, the domain is -5 ≤ x ≤ 5 and the range is also |y| ≤ 5.
     
    If you are given equations instead of graphs or tables, ask these two questions:
       1)  What values of x result in square rooting a negative number?
       2)  For what values of x is there division by zero?
    Rooting a negative? Dividing by 0?  
    Check for div by 0?  Check for square root of negative?
    Domain of h(x)  
     
    September 4th, 2012
    P-6 "Graphs of Functions"
    Function Graph
     
     Even Functions Odd Functions
     
    Greatest Integer [x]  
     
    September 6th, 2012
    "Function Transformations"
     
    Press Y= button.                        Press MATH button.                  Press right arrow to highlight "NUM" menu.
    Y=  Press MATH button  Hilight "NUM"ber menu
     
    Press 1 to select "abs(".               Press X key, then ")".            Press ZOOM key.
    select absolute value  Press X key. Press ZOOM key.
     
    Arrow to or press "6".               Press GRAPH key to graph Y1.      Press ZOOM and "6" keys.
    Select ZStandard  Zoom 6 Graph Press ZOOM and 5.
     
    to get the absolute value graph to square up like a "V".
    ZSquare 
    Enter these equations into calculator:
     Enter Y1-Y5 into calculator
    f(x) +2 moves f(x) = |x| up 2 units.   f(x+2) moves f(x) left 2 units.
    Y2 = Y1+2 Moves |x| up 2.  Y3 = Y1(x+2) Moves |x| left 2.
    How would f(x) get moved down?   How can f(x) be moved right 2 units?
     
     
    -f(x) reflects f(x) = |x+2| over x-axis.   f(-x) reflects f(x) = |-(x+2)| over y-axis.
    Y4 = -Y3 = -f(x) Reflects over x-axisY5 = -Y3 = Y1(-x-2) Reflects over y-axis
    Why are |x| and |-x| the same graph?
     
    Translating functions  
     
    A vertical stretch multiplies f(x)'s y-values, pulling it up.   A compression pushes it down by dividing.
    2f(x)  Vertical Stretch  f(x) / 2 Vertical Compression
     A horizontal compression pushes the "V" graph toward the center.  A horizontal stretch pulls it out to both sides.
    f(2x)  Horizontal Compression  f(x/2) Horizontal Stretch
    On the absolute value graph |x|, 2f(x) = 2|x| = |2||x| = |2x| = f(2x) so that the horizontal compression appears as a vertical stretch.
    Likewise for horizontal stretches and vertical compressions.  This is true for all linear functions. So let's change the parent function f(x).
     
    Can you identify each function's graph?
    Root (x) Parent function  Root x graphs
     
    Vertical Stretch = 4f(x)      Vertical Compress = f(x)/4   Horizontal Stretch = f(x/4)   Horizontal Compress= f(4x)         
    Vertical Stretch   Vertical Compress Horizontal Stretch Horizontal Compress
    Multiplying outside parentheses performs a vertical stretch, while dividing outside is a vertical compression.
    Multiplying inside parentheses is a horizontal compression, while dividing inside is a horizontal stretch.
    Note that multiplying inside () has the opposite effect horizontally than it does vertically.
    Vertical transformations match up with operations outside the ()'s.
    Horizontal transformations involve operations inside ()'s.
     
    Compression Formulas  
    Why are operations inside the parentheses opposite those from outside? 
    If y = f(x) + k, then y - k = f(x) moves the function down k units.
    Likewise, if y = k*f(x), then y/k = f(x) is a vertical stretch despite dividing by k.
     
     shift right
     

     Professor Pi approximating Pi with inscribed and circumscribed polygons using sine and tangent.
    Period 001   Dossier score
     
     

    Unit 4 Chapter 2
       Table of Contents
     
     
    Trig Identity Theft
     
    Lesson 2.1 Trig Identities
    Trigonometric Identities
     
    ID Categories   Quotient Identities
    Reciprocal Identities
     
    Even identity for cosine Cosine is an even function.
    Even ID for secant Secant is an even function.
    Odd Identities
    All four trig functions with sine in them are odd.
    wolowitz squared  
    Pythagorean Identity  cot^2 + 1 = csc^2
    sine^2 and cos^2 = 1  1 + tan^2 = sec^2
     Combining these three triangles into Quadrant I of the unit circle shows why ...
    Unit Circle Segments
    ...the tangent is named (it's the height of the segment that touches the circle at one point)
    and the secant (extend the purple hypotenuse to QIII where it will cross unit circle a second time.)
    Note here also that the cotangent and tangent segments are perpendicular
    (right angle90 degrees→complementary angles→tangent (complement) = cotangent(θ).)
    The cosecant is the hypotenuse of the red triangle, the secant the hypotenuse of the purple.
    The height of the brown triangle is the sine, and the base of it is the cosine.
    All six trig ratios are lengths of sides of these three triangles drawn onto the unit circle.
     EXTENSION:  So if θ is 30 degrees, or π/6 radians, find all six lengths.
     
    Lesson 2.4 Trig Formulas
     
    Lesson 2.4 Part I--Cosine Difference Formula
     cos(A-B) = cos(A)cos(B)-sin(A)sin(B)
    Lesson 2.4 Part II--Sum and Difference Formulas for Cosine(α-β) and  Sine (α±β)
     cos(A-B), sin(A+B), sin(A-B)
    tangent angle addition  
    tangent angle difference  
    Lesson 2.4 Part III--Double Angle Formulas for cos(2θ), sin(2θ), and tan(2θ)
    Cos(2x), Sin(2x), Tan(2x)  
     
    Lesson 2.4 Part IV--Half Angle Formulas for cos(x/2) and sin(x/2)
    Cos(x/2) and Sin(x/2)  
     
    Lesson 2.4 Part V--tan(x/2)
    tan(x/2)  
     
    Lesson 2.4 Part VI--Cotangent Formulas for Cot(α±β) and Cot(2θ)
    cot(a+b),cot(a-b), cot(2x)  
    sum to product
    Sum to Products KEY  
     

     
     
     Unit 3 Chapter 3.1-3.3 and 1.8
     
     
    Lesson 1.8 "Harmonic Motion"
    October 24th, 2011
     
    Lesson 1.8 "Solving Right Triangles"
    October 22nd, 2011
    Lesson 3.1B
    "Area of Triangle (SAS formula)"
    October 27th, 2011
     
    AREA = base * height / 2
    A = bc sin(A) / 2
     
    Lesson 3.1A
    "Law of Sines"
    October 25th 2011
     
     
     
     Lesson 3.2
    "Law of Cosines"
    October 31st, 2011

     
     
    Unit 2B Chapter 1.5-1.7 "Trig Graphs"
     
     
    Notes #0 (optional)
    September 27, 2011
    Graphing Calculator Keystrokes.
     
    At any time, press 2nd   MODE  to return to the calculator Home Screen. 
     
    Press   Y=  to enter equations to graph.  Press  GRAPH  to see the graph.
     
    Press  ZOOM  and  ): ZTrig to set the window to [-2π < x < 2π] by [-4 < y < 4],
     
    or press the down arrow cursor to "7" and press  ENTER .
      
    To enter unit circle values into a LIST, press   STAT   then ENTER.
    If there are any numbers in the columns, move the cursor to hilite L1, etc., and press CLEAR .
    Then place the cursor under L1 and type each number and ENTER in between each value.
    (i.e. where the commas are in this list {0, 30, 45, 60, 90, 135, 180, 270, 360})
     
    When done, quit to the Home Screen (see above.)  Enter the following keystrokes:
     2nd   1  2nd   / 180  ENTER, where "/" means divide. You should see "L1* π/180" on the screen.
    This converts all entries in list L1 to decimal radians. 2nd  is the π key. (π = pi)
    Press STO   2nd  to store this into list L2.  Or combine all steps:
     2nd  , 1 X , 2nd , ^,   / , 180 , STO , 2nd , 2 , ENTER  . You should see L1*π/180 -> L2 on your home screen.
     
    You can manually enter the unit circle y-values for sine into L3 as before but SIN(L1) STO  L3  ENTER  is faster.
    (2nd, 3 is L3). The first "(" is automatic but you have to enter the right "  )  ".
    The screen should read sin(L1) -> L2   followed by decimals in the brackets {0 ...} on the next line.
    Before moving on press 2nd, STAT, ENTER to confirm the entries in lists L1, L2, and L3.
     
    We're going to make a STAT PLOT of these values.  Press 2nd, Y=, ENTER  to go into the STAT PLOT menu.
    Hilite Plot1 and press ENTER, hilite "On" ENTER, then select the first type of stat plot (scatter).
    Arrow down to XList: and set it to L2 by pressing 2nd, 2. Select L3 for YList. 
    Leave the marker as the square (not the dot or cross.)
     
    Make sure your calculator is in radian mode by pressing MODE, hilite "Radian" or "Rad" and press ENTER.
    Then press  GRAPH  . You should see little squares on the graph.
    If you got a DIM: MISMATCH error, your L2 and L3 lists are of different sizes (you missed an entry.) Fix it and try again.
    If there are other lines on the graph you must CLEAR the equations in the Y= menu.
    Press Y=, CLEAR all equations, and enter sin(x) into Y1 by pressing  SIN  and  X,T,O, , and GRAPH. (O is supposed to be Theta)  If your graph does not appear, be sure the equation is turned on by hiliting the  sign after Y1.
    You can press WINDOW  to adjust the viewing settings of the graph. 
     
    Repeat the process for the Cosine on the back of the Sine and Cosine graphs worksheet.  Print this page for 1 point extra credit in Notes packet 2B and for future reference.
     
     

     
     
    Notes #5 (October 11th, 2011)
    First, enter these angles into List L1 and make these lists for sin, cos, and tan.
    calculator instructions
     
    Turn Plot1 "On" and graph (ZOOM 7) the sine graph for -90 < x < 180.
     sine plot inverse sine plot
    Then swap the Domain and Range with XList : L2 and YList: L1.
    Before graphing, set WINDOW to -4 < x < 4 by -360 < y < 360.  Then press GRAPH.
    The plotted points fail the vertical line test and so the inverse relation is not a function.  
    However, enter y = sin^(-1)(x) by pressing. Y=, 2nd, SIN, X to get the inverse sine function.
     
    inverse sin = ArcSine
    For cosine, we'll fist convert to radians, then repeat the process.
    cosine plot
     
    Again, the inverse cosine relation is not a function.  However,
    if we restrict the domain of the angle to run from 0 to π radians (180 degrees),
    we run through all the x-values on the unit circle for cosine.
     0 to pi domain and range range then domain ArcCosine graph
    This restricted cosine is called Cos(x) and its inverse function is ArcCos(x).
    (It's called an 'arc' because it returns the angle, or arc measure, on the unit circle.)
    -pi/2 to pi
     
     If the angle theta runs from 0 to 180 degrees (π radians), all the cosine values are covered from -1 to 1.
    To cover all the y-values of sine, the angle must go from -90 degrees to 90 degrees (or -π/2 to π/2 rad).
     Restricted Ranges swap domain and range
    For tangent, what angles must be covered so that the line through unit circle takes on every slope value?
     Restrict the domain to just graph one branch of tangent.  Call new function Tan(x) with capital T.
    y = Tan(x) on [-90 to 90]  y = ArcTan(x) on all Reals
    Then its inverse ArcTan(x) is graphed from interchanging the x- and y-values.  
    Domain of [-π/2, π/2] of Tan becomes Range of ArcTan and
    Range [-, ] of Tan becomes Domain of ArcTan.
    gets a tan  
     
     
     
     
    Notes #4 (October 10, 2011)
     
     Notes #4 intro to Inverses inverse notation   Now that we can find trig values of given angles, let's do it backwards.
    sin cos tan   so theta equals?
    This method of "undoing" what sin, cos, or tan does is called an "inverse."
    In the notation the "-1" means inverse, not an exponent for a reciprocal.
    (You wouldn't undo addition by division, and we won't undo sine by dividing for the same reason.
    They are different levels of inverses--add and subtract, multiply and divide, sine and inverse sine, etc.)
    inverse sine  inverse cosine & inverse tangent  inverse trig functions
    cos and inverse cos   and sin (3pi/4) yet inverse sine = pi/4
    sine (3pi/4) = sine (pi/4)   sine graph crosses 4x
    Both sin(π/4) and sin (π/4) have a value of 0.7071.  So what angle will inverse sine of 0.7071 output?
     
     
    Simplifying radicals with Mr. Happy Face!
    root happy Factor root*root happy/root(happy) rationalize denominator cancel and reduce
     
    Notes #3 (October 6, 2011)
     For using a graphing calculator with the tan-cot/sec-csc graphing worksheet:
    tangent and cotangent GRAPHS  secant and cosecant GRAPHS
    This is the graph of the tangent function.  y = tan(x)
    y = tan(x)  
    Note the intercepts where sin(x) = 0 and the asymptotes where cos(x) = 0.
    tan invalid undefined  
    What are these x-values?                                        To graph y = cot(x) enter 1/tan(x) or cos(x)/sin(x).
    asymptotes and intercepts  What to do with no "cot" key on calculator y = sec (x) ( = 1/cos(x))
     Label the x-intercepts and the asymptotes.  Where are they?
    Finish the cosecant graph on the worksheet.
     
    Let's recap the domain and range of all six trig functions:
    trig domains and ranges  
     
    Notes #2 (October 4, 2011)
     Phase Shifts
    Shifted left pi halves  
    sin(x+pi/2)  cos (x- pi/3)
    cos shifted  
    sin(2x-pi)  and its graph
    3cos(2x-pi/4)  scale its graph
     
    Notes #1 (October 3, 2011)
     Amplitude and Frequency
    Graph y = sin(x).  Then graph y = 2sin(x) with a vertical stretch factor of 2.
     sine waves
    Then graph y = -sin(x).  A stretch factor of -1 reflects over the x-axis.
    Notes 2B #1  vertical stretches Range of Stretches
    The original range of sin(x) of [-1, 1] is not affected by reflecting over the x-axis.
    However, 2sin(x) has a range of -2 <  y < 2.  What about y = -2sin(x)?
    Amplitude definition  y = -2sin(x)   
     The Amplitude is the absolute value of the stretch factor.  A = |a| if y = asin(x) and |-2| = 2.
     cosine amplitudes
    When dividing by 3, a vertical compression results the same as multiplying by 1/3.
    vertical compression   So the cosine graph will be sandwiched between y = 1/3 and y = -1/3.
    Now let's look at f(kx) , a horizontal compression factor.
    Horizontal Compression  Frequency/Factor of 2
    For y = cos(2x), the cosine cycled through its wave twice over 2π on the x-axis.
    Frequency Definition Normally, cosine has a Period of 2π, Period of sine/cosine , but here P = 2π/2 = π.
     
    If y = asin(bx), then Amplitude A = |a|, frequency = b, and its period = 2π/f.
     

    Unit 2A Chapter 1.1-1.4 "Trigonometry"
     

     
    Notes #6, Lesson 2-4
     
     September 26th, 2011
    Three rules for remembering unit circle coordinates:
    rules  
    Make a table of reference angles versus sine, cosine and tangent.
    ref angle table
    (Note: This table is transposed from the one in Unit Circle PDF and the notes below.)
    terminal side  
     
    In QII, the reference angle = 180 - θ or π - θ.
    In QIII, the formula for reference angle = θ - 180 or θ - π.
     
    definition  
     
    In QI, the reference angle is θ In QIV, it's 2π - θ or (360 - θ) for positive theta.
    For example, if θ = 330, then ref angle = 360-330 = 30 degrees.
    For a negative theta angle, simply take the absolute value:  e.g. |-45| = 45 degrees.
     
    example 1  
    In this case, we have a negative θ in QIII, so ref angle = 180 - |-135| = 45 degrees.
    Cosine is negative because so are the x-coordinates in Quadrant III.
    HW p155 2a  
     
    Period 3's Notes
    Reference Angle  Reference Angles
    This time, I color coordinated the table to be consistent with the Unit Circle from the Notes on 2.2.
    Degrees, Radians, Coordinates , with scratch work in purple and the table in brown and trig ratios in black.
    Trig Reference Table  
    Also, with this arrangement, the tan ratios of sin/cos are easy to generate because
    the sine values are actually right on top of the cosine values. Note the simple radical form
    for tan(30) as well as how the 1/2's cancel in tan(60).  And yes, root(3) = tan(60) = 1.732...
     
    Quadrants sincostan  
     
     
    Lesson 2.4 "Right Triangle Identities"
     September 24th, 2011
     
    Apply the Pythagorean Theorem to a triangle with
    sides cosine(θ), sine(θ), and hypotenuse of 1.
    Apply Pythagorus Pythagorean Identity #1  
     Then make a similar triangle by dividing all three sides by cos(θ) so the adjacent base is 1.
    Divide by cosine  Pythagorean Identity #2
    Divide all three sides of first triangle by sine(θ) so that height opposite angle theta is 1.
    Divide by sine  Pythagorean ID #3
    The five reference angles in degrees and radians for the three trig functions of sine, cosine, and tangent.
    Trig Table of Special Angles
     
     
     Lesson 2.3  "Trig Functions"
    September 22nd, 2011
     
    There are six trigonometric functions because there are six ways to choose two sides from three. 
    Since order matters in a ratio, we use a permutation of choosing k objects from n.
     
    Triangle Trig Functions  
    On an xy-grid, the three sides of the triangle become r = hypotenuse, y = opposite, x = adjacent and t = θ.
     triangle in circle Right Triangle
    Including the reciprocals of the three trig ratios of sine, cosine, and tangent from geometry provide the new ratios of cosecant (= csc), secant (= sec), and cotangent (= cot).
    Six Trig Functions
     Each of the three "new" trig functions is a reciprocal of one of the original three.  So tan = 1/cot.
    Quotient and Reciprocal IDs
     
    What ratio would result from cos/sin ?
     
    Lesson 2.2 "The Unit Circle"
     what's your sine
    Combining the 8- and 12-slice circles from the last lesson:
    Completed Unit Circle
     
    Since dividing the circle into eight and twelve sectors results in angles that are multiples of 30 and 45, we can fit the 30-60-90 and 45-45-90 triangles from geometry into the unit circle.  Scale each triangle so that each hypotenuse is 1 unit in length.  This means dividing by the original hypotenuse of 2 or √2.
    special unit right triangles  
     
    Convert each triangle leg into simple radical form by rationalizing each denominator.
    As needed, multiply the top and bottom of the fraction by √2.
    Then the coordinates (x,y) are obtained by imbedding special right triangles into the circle:
     
    Special Right Triangles
     
    triangle 60
    triangles across axes  
    and inserting + and - signs for x- and y-coordinates depending on the Quadrant:
    Quadrant II (-, +) Quadriant I (+,+)
     
    Quadrant III (-,-) Quadrant IV (+,-)
    to get the completed unit circle.
     

     
    Lesson 2.1 "Radians and Degrees" 

    September 12th, 2011
     
    First divide a circle of radius 1 centered on the origin into 8 equal slices.  List each degree measure:
    Quadrant I Eight slices Eight slices of circle
    One complete revolution is both 2 pi radians and 360 degrees.  So the conversion factor is:
    2pi/360 = pi/180 to go from degrees to radians. Multiply by 180/pi to convert back to degrees.  
    360 degrees = 2pi radians
     
    Divide another unit circle into twelve equal slices.  The first quadrant (QI) will look like:
    QI of twelve slices 12 radian slices of unit circle
    And the whole thing looks like
    Converting degrees to radians
     
     Both 8- and 12-slice circles
    HW:  Combine both circles onto one giant circle so we can find the (x,y) ordered pair coordinates.
     
     Checking the conversion factor:
     Converting deg to rad
    Period 3's Notes
    Eighths
    Quadrant I of 8 slices Quadrant II of 8 slices QIII of eight slices QIV of 8 slices
     
    Eighth radians 2 pi radians = 360 degrees
     
    Twelfths
    Period 3's twelve slices
     
     too damn pi

    Unit 1 Chapter "Prerequisites"
     Lesson P.9 "Inverses" (Notes #9)
     Domain and Range
     
    Graph of Celsius-Fahrenheit inverses
     
     Proving Inverses fog and gof

     
    Lesson P.8 "Function Compositions"  (Notes #8)
    Notes 6
    Composition Notations
    (f+g)2 and (f-g)(2)
    (f*g)(2) and (f/g)(2)
    (g-f)(2) and (g/f)(2)
    Operations w/ f(x) and g(x)
    fog(x) and gof(x) and of 2
    gof(x) and gof(2)
     
     
     


     
    Unit 4 Chapter 3 "Solving Triangles"

    STAR TRIG
    Lesson 3-1 "Law of Sines" (11/23/2010)

    Lesson 3-2 "Law of Cosines" (11/30/2010)
     
     

     

Last Modified on November 28, 2018