ࡱ> b .jbjb =mmE<4- 6666")))(*+"-----===,R=*===}66--wt}}}=6 --}$X6666=}}6}z- 90) Based on Sides Equilateral (all sides =) Isosceles (2 sides =) Scalene (all side `") Similar " s Congruent ! Similar (not reverse) " s = & Sides in proportion Called Correspondence Corresponding " s = Corresponding sides proportional ratios = Vertical " s Angles across from one another, formed when two lines intersect.  Parallel Lines Cut By a Transversal Corresponding " s  Same side of transversal & ll lines  Alternate Interior " s  Inside of ll lines & alternate sides of transversal   Alternate Exterior " s  Outside of ll lines &  alternate sides of transversal Example: Find the measure of " s 1,2,3&4  Triangles A geometric figure (polygon) with three sides. The sum of angles in a triangle is 180.    Example: Find "B. Types of" s Based on Sides Scalene  All sides different lengths Isosceles  Two sides same lengths Equilateral  All sides same lengths  Types of " s Based on "'s Acute  All " s less than 90 Right  One " is 90 Obtuse  One " is greater than 90  Example: Classify each by side and " s a) b) c) 1.3 Trigonometric Functions Outline Standard Position x, y & r r = "x2 + y2 6 Trigonometric Functions sine, cosine, tangent, cotangent, secant & cosecant Exact Values vs Approximate Values Exact: Std. Form, x, y & r or Right Triangle opposite, adjacent & hypotenuse Approximate: Calculator in DEG mode Quadrant " s Unit Circle Drawing w/ Calculator Table p. 27 Standard Position of "q  Based on the "q in Standard Position the 6 trigonometric functions can be defined. The names of the 6 functions are sine, cosine, tangent, cotangent, secant and cosecant. Because there are many relationships that exist between the 6 trig f(n) you should get in a habit of thinking about them in a specific order. Ive gotten used to the following order and Ill show you some of the important links. Note 1: These are the exact values for the 6 trig f(n). A calculator will yield only the approximate values of the functions. Note 2: This is both the definitions from p. 23, their reciprocal identities on p. 30 and relations that dont tie to the coordinate system (an " in standard position). The definitions given in terms of opposite, adjacent and hypotenuse will help in Ch. 2. How to Find the Exact Values of the 6 Trig F(n) Step 1: Draw the " in the coordinate system creating a " w/ the terminal side and the x- axis. Step 2: Place , x & y and find r (using r = "x2 + y2) Step 3: Use x, y & r w/ definitions to write the exact values of the 6 trig f(n) Note: Once you ve got sin, cos & tan you ve got the others due to the reciprocal identitites. Step 4: Simplify (you ll need to review your radicals) Example: The terminal side of  in standard position passes through the point given. Find the value for the 6 trig f(n) of each " created by the ray passing through the origin and the given point P. a) (12, 5) b) (8, -6) It is important to note that the 6 trig f(n) values change depending on  s measure, not upon the point through which r passes. This is because along every ray, there are an infinite number of ordered pairs. Or, said another way, There are many ordered pairs that will create the same ray. Example: For the ray OP shown above, compute sin, cos and tan using P2 (4, -3). Do you see that these 3 (and therefore also cot, sec & csc b/c of the reciprocal identitities) are identical to those computed in the part b of the last example? Note: What do you notice about the slope of the ray OP and the tan ? In general they will always be the same. Now we can extend this idea to the fact that any ray can be described by: Ax + By = 0 where x e" 0 or x d" 0 Thus we can use a ray s equation to find the trig f(n) for "q formed by the terminal side by such a ray. Finding 6 Trig F(n) Given a Ray s Equation Step 1: Let x = value in the domain and solve for y (finding an ordered pair on the ray) Step 2: Draw the ray with x&y from step 1 and create the right triangle. Step 3: Find r using x & y Step 4: Use the definitions for the 6 trig f(n) to find their exact values Example: Find the exact values fo the 6 Trig F(n) for "q in standard position if the terminal side is defined by 3x  2y = 0 and x < 0. Let s do an example that emphasizes critical thinking and knowing the 6 trig f(n) *Example: Decide if the function or ratio would be positive or negative. [Hint: Think (x, y) values in Quadrants] a) I, sin  b) II, y/x c) III, cot  d) IV, r/y Quadrant " s  The " s that lie on the axes.  Quandrant " s on x-axis -- 0, 180 & 360 (or their multiples) X = 1 Y = 0 R = 1 Quandrant " s on y-axis -- 90 & 270 (or their multiples) X = 0 Y = 1 R = 1 Function Values of Quadrant (s (Table p. 27 Just Keep It Close) (sin (cos (tan (cot (sec (csc (0(010Undefined1Undefined90(10Undefined0Undefined1180(0-10Undefined-1Undefined270(-10Undefined0Undefined-1360(010Undefined1Undefined Example: Find the trig f(n) values for each " a)  = 360 b) "q w/ terminal side thru (0, -5) [Hint: Remember the example w/ similar " s on p. 11of notes. It doesn t matter if it s (0, -1), (0, -2) or (0, -5) all are same ray] Example: Decide if the f(n) value is 0, 1, -1or undefined. Note: n" 180 is an even multiple of 180 and (2n + 1)" 90 is an odd multiple of 90 a) tan [(2n + 1)" 90] *b) cos [270 + n " 180] c) sin [n " 180] Calculator Notes: Example: Use calculator to find the approximate values of sin , cos  & tan .  = 90. Note1: Make sure you are in DEG mode [MODE ! DEG not RAD] Note2: tan  for 90 is undefined. Note3: Finding the sin 90 is always a good starting place to assure you are in the correct mode, since it should always yield 1. *Example: Let s draw a unit circle w/ our calculators. a) Set to DEG & PAR then 2ND!MODE to quit  b) x1T = cos ( or  y1T = sin ( or  c)  d)  e) Make sure x1T = cos (T) and y1T = sin (T) are at top  or to move around the circle T = Degrees X = Value of cos (T) & Y = value of sin (T) Example: Move to xH" 0.766 and y H"0.643 What is the value of T for cos (T) H" 0.766 (in QI)? For what value of T are cos & sin T H" 0.7071? For what value of T are cos T H" -0.866 (in QII)? Note: This is an idea that we will study in the next chapter called inverse trig f(n) / 1.4 Using Definitions of Trigonometric Functions Outline Reciprocal Identities (Needed in Calculus) Review Caution: Not inverse f(n) Different Forms Since 1/sin = csc then sin csc = 1 Using vs Strict Definition Signs & Ranges Relate to Quadrants All Students Take Calculus p. 31 Table Sign based on Quadrant Ranges Think of the unit circle sin & cos Cot & Tan based on sin & cos Sec & CSc based on 1/sin & 1/cos Trig F(n) Review & Extend 1.3 Pythagorean Identities (Needed in Calculus) Developed on p. 34 Sin2  + cos2  = 1 or cos2  = 1  sin2  Tan2  + 1 = sec2  1 + cot2  = csc2  Quotient Identities (Review  Also needed in Calculus) Sin /cos  = tan  not for 90 & 270 or multiples Cos /sin  = cot  not for 0, 180 & 360 or multiples Reciprocal Identities (Important in Calculus) These come from the definitions and they are just the pairings that relate to being the reciprocal of the given trig f(n)  sin , cos  or tan .  Remember fo course that for 0, 180 & 360, y= 0 and therefore csc  & cot  are undefined and for 90 & 270 x = 0 and sec  & tan  are undefined (see p. 27 table). Note: sin-1 on a calculator is not 1/sin . Inverse functions wait until 2.3 for discussion, so more later.  Note: This is true for all the Reciprocal Identities. Example: Find each f(n) value given the information a) tan , given cot  = 4 b) sec , given cos  =-2/ "20 c) sin , given csc  = "18/2 Signs & Ranges of Function Values You don t have to memorize this, but you at least have to be able to develop it, which is dependent upon knowing quadrant information and standard position.  Lets go through the QII information using the definitions of the 6 trig f(n) to see how this works: In QII, x is negative (x < 0) while y & r are positive (y, r > 0) So, sin = y = + = + csc = r = + = + r + y + cos = x = = sec = r = + = r + x tan = y = + = + cot = x = = + x y + So, you can always develop the table on p. 31 (see below), memorize it or use All Students Take Calculus to help fill in the table. ( in Quadsin (cos (tan (cot (sec (csc (I++++++II+----+III--++--IV-+--+- Example: Determine the signs of the trig f(n) of the following " s. [Hint: Think quadrants! Remember coterminal " s for part c.] Along these same lines let s look at determining which quadrant an angle will be located in based on the sign of the functions. Example: Identify the quadrant (or quadrants of "q satisfying the given condition. [Hint: Go back to definition & use quadrant information.] a) tan  > 0 & csc  < 0 b) sin  > 0 & csc  > 0 Ranges of Trigonometric Functions Note: Every "q between will have values between -1 & 1 as x & y will never be larger than +1 nor smaller than -1. So, Since tan  = y and sometimes x < y, x = y and x > y x  Since csc  & sec  are reciprocal f(n) of the sin & cos (respectively)  This information is contained, in a different form in the table on p. 33. Example: Decide whether each is possible or impossible. a) cot  = -0.999 b) cos  = -1.7 c) csc  = 0 Now we will combine this with finding all the f(n) values based on quadrants and signs associated with those quadrants. Example: Suppose "q is in QIII and tan  =8/5. Find the other 5 trig f(n) values. [Hint: + = +/+ or  / ] On p. 34 of your book you ll find a discussion of the identities  exercise 81 should solidify the explanation. Pythagorean Identities (Very Important for Calculus)  For a cool explanation go to: http://www.coolmath.com/lesson-pythagorean-identities-1.htm Quotient Identities (Very Important for Calculus) You already know one of these, because I talked about it along with the definitions of the 6 trig f(n). The other is just its reciprocal. sin  = tan  cos  = cot  cos  sin  Y. Butterworth Ch. 1 Notes  Lial Trig  PAGE 1 Initial Side Terminal Side ( y x Origin = Vertex QIV QIII QII QI (3r) (2r) (7t + 4) (2t + 5) 1 2 3 4 "1 = 2 3 = 4 A B C D F E G H "A =E B=F "D =H C=G "C =E B=H "A =G D=F Interior " s On Same Side of Transversal "C + H = 180 B + E = 180 "3 = (7x " 5) "2 = (9x + 9) "4 "1 A C B "A + B + C = 180 B C A 48 61 B B A A C C B A C A B C A B C A B C C B = 135 A 8 8 r = ( x2 + y2 (Pythagorean Theorem) r > 0 since it is a distance (an undirected vector meaning it has no direction) r is hypotenuse x is adjacent 60 60 y is opposite ( P (x, y) sin ( = opp = y hyp r cos ( = adj = x hyp r tan ( = opp = sin ( = y x ( 0 adj cos ( x cot ( = adj = 1 = cos ( = x y ( 0 opp tan ( sin ( y sec ( = hyp = 1 = r x ( 0 adj cos ( x csc ( = hyp = 1 = r y ( 0 opp sin ( y P2 (4, -3)  P (8, -6) x y O The ray OP is a portion of the line y = -3/4 x and 4" P2 also lies on the ray OP. Both points will yield the same values of the 6 trig f(n). 0, 90, 180, 270 & 360 and their multiples x + n " 360 (0, 1) x = 0 y = 1 r = 1 (-1, 0) x = -1 y = 0 r = 1 x = 0 y = -1 r = 1 (0, -1) (1, 0) x = 1 y = 0 r = 1 Y = 0 in Numerator F(n) = 0 Sin & Tan Y = 0 in Denominator F(n) = undefined Csc & Cot X = 0 in Numerator F(n) = 0 Cos & Cot X = 0 in Denominator F(n) = undefined Sec & Tan Note: This begins the concept of the unit circle which will be studied in Ch. 3. MODE y = ALPHA 4 X, T, , n X, T, , n 4 ALPHA WINDOW GRAPH TRACE ! ! ! sin  = 1 & csc  = 1 csc  sin  cos  = 1 & sec  = 1 sec  cos  tan  = 1 & cot  = 1 cot  tan  Equivalent Forms Since sin  = 1 csc  & csc  = 1 sin  This means that (sin )(csc ) = 1 Tan & cot  + Cos & sec  + Sin & csc + All F(n) + y x QIV x & r > 0 , y < 0 QIII x & y < 0 , r > 0 QII x < 0, y & r > 0 QI x, y & r > 0 This Saying Will Help Remember the Positive F(n) All All f(n) + Students sin & csc + Take tan & cot + Calculus cos & sec + Max & Min Values of Sine & Cosine are 1 & -1 sin 0 = sin 180 = sin 360 = 0 = 0 1 sin 90 = 1 = 1 sin 270 = -1 = -1 1 1 cos 0 = cos 360 = 1 = 1 1 cos 90 = cos 270 = 0 = 0 1 cos 180 = -1 = -1 1 (0, -1) (-1, 0) (0, 1) (1, 0) -1 d" sin , cos  d" 1 -" d" tan , cot  d" " -" < csc , sec  d" -1 or 1 d" csc , sec  < " sin2  + cos2  = 1 or cos2  = 1  sin2  tan2  + 1 = sec2  1 + cot2  = csc2  DFHRT   " d l H J X ( F   \ ^ r ~ )6ڤ{p{hFmhS OJQJ#jhS UaJmHnHsHtH,jhFmhS 5UaJmHnHsHtH&jhS 5UaJmHnHsHtHhFmhS 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