Principles of Mathematics Section, Introduction 5 Module, Section Graphs of Trigonometric Functions Introduction You have studied trigonometric ratios since Grade 9 Mathematics. In this module ou will stud the ratios as functions. You will look at tables of values of these functions, generate their graphs, and list their properties. It is important for ou to do all the questions in the assignments because some new concepts are developed within the assignments. Using techniques ou developed in Module, ou will stud transformations of these circular functions. In order to find the zeros of the functions, ou will develop skills in circular function equation solving and in relationships amongst these functions which are called identities. Section Outline Lesson Lesson Lesson 3 Review Sine and Cosine Graphs Transformations of the Sine and Cosine Functions Graphs of the Remaining Circular Functions Module
Section, Introduction Principles of Mathematics Notes Module
Principles of Mathematics Section, Lesson 7 Lesson Sine and Cosine Graphs Outcomes Upon completing this lesson, ou will be able to: sketch the basic graphs of = sin and = cos list the properties of each of the above two functions sketch transformations of the above two functions and describe how the basic properties have been altered solve unrestricted sine and cosine equations Overview Now that ou can find circular function values for an real number ou are read to sketch the graphs of these functions. The Sine Curve If ou do not have a graphing calculator, use a pen or pencil to finish the diagram below, recording our observations as the central angle θ moves from quadrant to quadrant. The idea is to get a feel for the gradual rise and fall of a -value as it travels around a circle. θ 0 3 Keep in mind what ou learned in the last section: on the unit circle, a -value for a point is the same as a sin θ value for the central angle. Module
8 Section, Lesson Principles of Mathematics as θ increases -values sin θ curve In Quadrant I, are positive and starting at the origin, increasing from the curve rises slowl from 0 to 0 to from 0 to, at θ = In Quadrant II, from to In Quadrant III, from to 3 are positive and decreasing from to 0 are negative and decreasing from 0 to the curve declines slowl from to 0, at θ = the curve declines slowl from 0 to, at θ = 3 In Quadrant IV, 3 from to are negative and increasing from to 0 the curve rises slowl from to 0, at θ = The result is a portion of the sine curve, = sin θ. As the choice for the independent variable is arbitrar, ou will often be asked to sketch = sin instead of = sin θ. It reall does not matter; it s simpl a choice of names. 3 θ Module
Principles of Mathematics Section, Lesson 9 On our graphing calculator, ou can sketch the curve b following these steps:. Be sure our calculator is in radian mode.. Set our viewing window as follows: X min = 0. X ma = 7 X scl = Y min =. Y ma =. Y scl = 3. Press Y= and enter sin(x) as Y. 4. Press GRAPH. You will get the following graph: B zooming 7:ZTrig ou will see the curve begin at θ =.585... and end at θ =.585 (You can confirm these values b checking the X min and X ma values b pressing WINDOW). The curve will go through almost two ccles. Etra for Eperts You can do a simulation to see how the sine curve evolves from the unit circle b plotting the unit circle and the sine curve parametricall. Follow these steps:. Press MODE and select Par instead of Func.. Press Y= and enter the unit circle as follows: X T = cos(t) Y T = sin(t) These are the coordinates on the unit circle (cos θ, sin θ). Module
70 Section, Lesson Principles of Mathematics 3. On the same screen, enter: X T = T Y T = sin(t) Note: This is the equation = sin θ. 4. Change our viewing window as follows: T min = 0 T ma = 7 T step = 0. X min =. X ma = 7 X scl = Y min =.7 Y ma =.7 Y scl = 5. Press GRAPH and watch as the unit circle is drawn.. TRACE both curves and ou will see how the -values (i.e., sin θ values) are related to the unit circle s -coordinates. 7. Tr zooming ZSquare and ZTrig. TRACE both curves again to get a better feeling for the sine curve and its evolution. Properties of Sine Curve You should be able to draw a sketch of the sine curve without an aids and list the properties stated below.. The domain is the set of all real numbers, since ou can travel around the unit circle as far as ou wish, clockwise or counterclockwise. Make sure ou understand that the unit circle is used to generate the coordinates (cos θ, sin θ). However, the domain of sin θ and cos θ is similar to a string wrapped around the unit circle. This is the reason that these functions are sometimes referred to as wrapping functions.. The range is [, ]. 3. The -intercept is 0. Module
Principles of Mathematics Section, Lesson 7 4. The zeros of the function are all integral multiples of, that is,...,, 0,,, 3.... A more concise epression of the function s zeros is the notation {k k I}. Notice that the solution is not restricted to an interval such as [0, ]; therefore, the solution is infinite and we give the solution set as a formula to generate all the possible zeros. 3 3 4 θ Functions with a repeating pattern like this sine function are called periodic functions. Eamples of periodic functions are all around ou: tides, the motion of a Ferris wheel, and so on. The time for the pattern to repeat itself is called the period. For patterns in wallpaper or in a cable sweater, the length between each pattern repetition is the period. Mathematicall a period p can be a measure of length or time; after that period p, or interval p, the function repeats. Definition: A function f() is a periodic function if there eists a number p > 0, such that for all in the domain of f, f( + p) = f(). The smallest such number p is called the period of f. (The period is the shortest distance ou must travel along the -ais for the function to begin another ccle.) 5. The period for the sine function is. θ Module
7 Section, Lesson Principles of Mathematics. The graph below shows that sin () is smmetric about the origin. On an line drawn through the origin that intersects the graph, the origin is the midpoint (the smmetr point) of that straight line. When that is true, it follows that sin ( ) = sin() or sin() = -sin( ). Smmetr about the origin is called odd smmetr. We'll eplain wh it's odd after ou learn about the other kind ( even smmetr) in the net Guided Practice eercise. θ 7. Definition: Curves with wavelike forms, such as the sine curve, have a line midwa between the high and low points of the curve. This line is called the ais of the curve. The distance from the ais to the maimum, or to the minimum, curve values is called the amplitude of the curve. The amplitude of the sine curve is. θ Module
Principles of Mathematics Section, Lesson 73 If we restrict our attention to one ccle of the sine curve, from 0 to, and subdivide the curve into the four quadrants, two additional properties are apparent (see 8 and 9). I II III IV θ 8. Since the curve is above the ais in Quadrants I and II, it follows that sin θ > 0 in these two quadrants. Similarl, sin θ < 0 in Quadrants III and IV. This agrees with the CAST rule. 9. As ou move along the curve from left to right the curve is increasing (moving up) in Quadrants I and IV and decreasing in Quadrants II and III. Module
74 Section, Lesson Principles of Mathematics Guided Practice The Cosine Curve Sketch the curve = cos θ using the unit circle or graphing calculator. List the above nine properties of the cosine curve as indicated below.. Domain. Range 3. -intercept 4. Zeros 5. Period. Smmetr: Does cos() = cos( )? or does cos() = cos( )? 7. Amplitude 8. In one revolution, when is cos θ a) positive? b) negative? 9. In one revolution, when is cos θ a) increasing? b) decreasing? Check our answers in the Module Answer Ke. Module
Principles of Mathematics Section, Lesson 75 Lesson Transformations of the Sine and Cosine Functions Now that ou know how to draw the basic sine and cosine curves, ou will turn our attention to some transformations of these basic curves. Eample Sketch the graph of: a) = 3 sin and state its range and period b) = cos and state the values of its intercepts Solution a) Notice that ou are graphing against. The choice of or θ letters is arbitrar. Just do not confuse this independent variable with the dependent variable on the unit circle. There is no connection between them. 3 3 3 = 3 sin stretches the sin graph verticall b a factor of 3. Range = [ 3, 3]. Period is still. Module
7 Section, Lesson Principles of Mathematics b) Reflects the graph over the -ais. -intercept =. -intercepts are still Eample Sketch and state the period of: a) = sin 4 F b) = cos θ (Recall: The period is the length of ccle of the graph.) Solution a) HG I K J ( k + ) k I. odd multiples of 3 Period is = because the curve is compressed b a factor of 4. 4 Module
Principles of Mathematics Section, Lesson 77 A useful hint: Man students find it easier to sketch circular function curves b using the normal period of the function to determine where the transformed curve begins and ends one ccle. For eample, the period of a basic sine curve is ; therefore, find the starting and ending point of one ccle as follows: Curve Start one ccle at End one ccle at Period basic = sin 0 transformed function = sin 4 When does 4 = 0? When = 0 When does 4 =? When = basic = cos 0 transformed function = cos When does = 0? When = 0 When does? = When = 4 4 b) Shifts the graph units to the right. To illustrate the use of the hint, determine the starting and ending value of one ccle. Start End F I θ HG K J = 0 θ = θ = 4 θ = + 5 θ =
78 Section, Lesson Principles of Mathematics Begin drawing the basic ccle of the cosine curve at θ = and finish the ccle at θ = 5. The horizontal translation of shift of θ =. to the right is also called a phase Note: We ll define phase shift more full in Section 4 Lesson. 5 start finish 5 Period is still =. Note: With eperience ou will notice that the period is onl affected if ou change the numerical coefficient before the (or θ). Eample 3 The graph below is a representation of the function = a sin b( c). Find a, b, and c. 5 5 Module
Principles of Mathematics Section, Lesson 79 Solution B inspection a = 5 period = = b = b phase shift = c = Eample 4 Sketch at least one ccle of the graph of: a) = sin 3 + Compare to the general form = A sin [B( C)] + D A = ; B = 3; C = ; D = 0 From A =, the base graph of = sin is reflected in the -ais and the amplitude is changed to. From B = 3, the period of the graph is =. B 3 From C =, the phase shift is or to the left. From D = 0, there is no vertical displacement. The graph is most easil sketched in stages:. Vertical shift is zero, so the graph lies along the -ais.. The amplitude is, so lightl sketch a straight line units up from the ais and units down, i.e., through = and =. 3. The phase shift is, so the graph will begin units to the left. Mark an X at,0. 4. The period is, so mark points on the -ais in 3 multiples of from. 3 Module
80 Section, Lesson Principles of Mathematics e.g. + = 3 5 + = 3 5 9 3 + = = 3 And to the left. 7 = 3 7 = 3 So far our set-up should look like this : 7 5 9 5. Each period has four major points, alwas in the same order, for either a sine or cosine curve: zero, maimum, zero, minimum. In the case of a sine curve with a negative amplitude, the order is zero, minimum, zero, maimum. Divide each period into four equal sections, starting at, and place an at the four major points of each period. Check that there are s in each of the zeroes ou calculated in step 4. 7 5 9
Principles of Mathematics Section, Lesson 8. Join the points with a smooth curve. 7 5 9 b) = cos 3 Note that this function is NOT in the form = A cos [B( C)] + D First, we have to remove the coefficient of from inside the brackets b taking out a factor of. We do this b rewriting as or to crea te a common 3 factor of. Function is = cos or = cos Now A = amplitude = [ no change] B = period = = C = phase shift is to the right D = vertical displacement is i.e., mid-line or ais for the graph is =. Vertical shift is, so the ais lies on the line =.. The amplitude is, so the maimum and minimum points go through lines one unit up and down, respectivel, from, i.e., = 0 and =. Module
8 Section, Lesson Principles of Mathematics 3. The phase shift is, so the graph will begin units to the right. 4. The period is, so mark points on the graph in multiples 7 3 5 of, starting at i.e.,,, and left at,. 5. The cosine pattern is ma, zero, min, zero. Divide each period into four equal sections and mark in the major points. 5 7 3. Join the points with a smooth curve. 5 7 3 Module
Principles of Mathematics Section, Lesson 83 Guided Practice. Sketch each of the following. State the domain, range, amplitude, -intercept, and period. a) = sin b) = sin c) = sin( ) d) = cos e) = cos f) = cos + 4 g) = sin h) = 3sin 3 i) = sin 3 j) = cos k) = cos l) = sin. For each of the following graphs of = a sin b( + c), find the values of a, b, and c. a) b) 3 5 3 3 Module
84 Section, Lesson Principles of Mathematics 3. Use the same graphs as in Question, but change the function to = a cos b( + c) and find the values of a, b, and c. 4. Find the period of each function. a) = 3 sin b) = sinb g c) = cos d) = sin F 3 4 I e) = cosb4+ g f) = sin HG K J b g b g 5. Find the -intercepts of each circular function. a) = sin 3 b) = cos c) = sin d) = sin e) = cos f) = 3 sin Etra for Eperts. Sketch = cos cos. 7. Sketch = sin + sin. ( ) Check our answers in the Module Answer Ke. Module
Principles of Mathematics Section, Lesson 3 85 Lesson 3 Graphs of the Remaining Circular Functions Outcomes Upon completing this lesson, ou will be able to: sketch the graph of = tan θ list the properties of the function = tan sketch the basic graphs of the reciprocal functions: = sec, = csc, and = cot list the properties of each of the above reciprocal functions Reciprocal Functions Eample : Sketching = tan θ For angles terminating in Quadrant I, tan θ begins ver small (infinitel close to 0) and becomes infinitel large as θ gets sin closer and closer to. Of course, since cos = 0, tan = = cos 0 which is undefined. Therefore, we see an asmptote at θ =. θ Module
8 Section, Lesson 3 Principles of Mathematics In Quadrant II, for values of θ ver close to but greater than like.58 r,.59 r, etc., value of tan θ start as ver large negatives (tan.58 r = 08.5, tan.59 r = 5.07). As θ gets closer to, the values for tan θ approach zero (tan 3. = 0.03, tan 3.3 = 0.0, tan = 0)., θ In Quadrants III and IV, the curve repeats its performances in Quadrants I and II, respectivel. 3 θ 3 θ Etra for Eperts You can plot the tangent curve b following these steps:. Press MODE and select Func.. Enter Y = tan(x). 3. Press GRAPH, then Zoom ZTrig. 4. TRACE the branches of the curve. Module
Principles of Mathematics Section, Lesson 3 87 We can observe and record the following properties of the function = tan : Domain: Range: R -intercept: 0 Zeros occur at: 0,,,,, etc.... Equations of asmptotes: Period: Smmetr: tan( ) = tan(), or smmetric with respect to (0,0). Behaviour: Increasing in all quadrants Positive in Quadrants I and III. Negative in Quadrants II and IV Eample : Reciprocal Functions Sketching = sec We are going to graph = sec b using the skills acquired in Module, Section, Lesson 5. To sketch R, k+, k Ι The set of all real numbers which are not odd multiples of. = sec = cos Step : Sketch = cos { R /, = k, k Ι} ( ) 3 =, =,..., = ( k+ ), k Ι Module
88 Section, Lesson 3 Principles of Mathematics Step : We now need to graph the reciprocal of this function. Since an intercept ( = 0) has no reciprocal, it follows that the vertical asmptotes are at: (, ),,0 (, undefined ) 3 3,0 (, undefined ) ( n ) 3 5 + =±, ±, ±,... = where n are integers Step 3: Plot the invariant points which occur when = and = Module
Step 4: Plot a few points b taking the reciprocal of the values. Notice that ver small positive and negative values, found on either side of the asmptotes, become ver large positive and ver large negative values respectivel. Step 5: Complete the sketch b joining these points with smooth curves. Step : The characteristics of the graph = sec are ( ) ( ) Domain:, Range: or Asmptotes: where are integers n n n + R ± + = ( ) ± ± ± = ± ± ± = ± ±, 3, 3.44, 4, 4, 4.55, 3, 3,,, Principles of Mathematics Section, Lesson 3 89 Module
90 Section, Lesson 3 Principles of Mathematics Eample 3: Graph = csc Step : To graph = csc = ( ) = sin sin, graph = sin first. Step : We now need to graph the reciprocal of this function. Since an intercept ( = 0) has no reciprocal, it follows that the vertical asmptotes are at: =±, ±, ± 3,... = n where n are integers Step 3: Find the invariant points which occur when = ± Module
Principles of Mathematics Section, Lesson 3 9 Step 4: Plot a few points b taking the reciprocal of the values and sketch a smooth curve.,, ( ) Step 5: Verticall epand the graph b a factor of. Step : Notice the characteristics. Domain: R, n where n are integers Range: or Asmptotes: = n where n are integers Module
9 Section, Lesson 3 Principles of Mathematics Guided Practice. Each of the remaining functions is a reciprocal of one of the basic three circular functions. Use the graphs of the basic functions to sketch their reciprocal functions: = csc, = sec, and = cot.. For each of the three functions in Question, list all the properties of each function: Domain Range -intercept Zeros Equations of asmptotes Periods Smmetr (odd or even) Behaviour 3. Sketch each of the following transformations: a) = tan b) = sec + c) = csc d) = cot e) = tan f) = sec Note: You ma wish to draw the reciprocal functions on the calculator b sketching: = = = sin,, and cos tan Check our answers in the Module Answer Ke. Review Section before attempting the review questions following the summar on the net page. These questions should help ou consolidate our knowledge as ou prepare Section Assignment.. Module
Principles of Mathematics Section, Summar 93 Summar of Section Know the basic shapes of all trigonometric functions. = sin period = amplitude = domain =R [ ] range =, = cos period = amplitude = domain =R [ ] range =, = tan period = asmptotes at ( k + ), k I ( k + ) domain =, k I range = R = cot period = asmptotes at k, k I { k k I} domain =, range = R Module
94 Section, Summar Principles of Mathematics = csc period = asmptotes at k, k I { k k I} ( ) domain =, range =, ] [, = sec period = ( k + ) asmptotes at, k ( k + ) domain = ( ) range =, ] [, Transformations of Sine and Cosine ( ) cos ( ) = Asin B C + D or = A B C + D domain = R [ A+ D A+ D] range =, amplitude = A period = B phase = C (units left) ais at = D Module
Principles of Mathematics Section, Review 95 Module, Section Review. Find the period of each function. a) = 4 sin b) = sin ( ) ( ) c) = cos 3. Sketch each of the following. State the domain, range, amplitude, -intercept, and period. a) = sin b) = cos ( ) c) = sin + d) = sin ( ) e) = sin f) = cos g) = sin h) = sin i) = cos j) = tan k) = sec l) = csc 3. Sketch at least one period of: ( ) a) = 3sin + ( ) b) = 5cos + + 3 For each graph determine: i) the domain ii) the range iii) the period iv) the amplitude v) the phase shift vi) the -intercept 4 Module
9 Section, Review Principles of Mathematics 4. Sketch the graphs of the following. Do our graphs look familiar? Hint : Rewrite as a) = sin b) = cos Check our answers in the Module Answer Ke. Now do the section assignment which follows this section. When it is complete, send it in for marking. Module
Principles of Mathematics Section Assignment. 97 PRINCIPLES OF MATHEMATICS Section Assignment. Version 05 Module
98 Section Assignment. Principles of Mathematics Module
Principles of Mathematics Section Assignment. 99 Section Assignment. Graphs of Trigonometric Functions Total Value: 40 marks (Mark values in margins) (8 marks). Fill in the blanks with the correct response. a) The figure below shows a sketch of the graph of = sin 3. The -coordinate of the point P in the diagram is: P b) The period of function f is 5. If f () = 4, f () = 5, and f (4) =, the value of f (7) is: c) The phase of the function f() = 5 cos ( + ) is: d) The range of the function = 3 sec is: e) The -intercept of the function g() = sin ( ) + is: f) The range of the function f( ) = 4sin is: 3 g) The domain of the function = tan θ is: h) Find the equations of the asmptotes for = csc(3θ): Module
00 Section Assignment. Principles of Mathematics. Sketch the graph of f() = 3 sin and state the period and range of this function. (4) f() Period: Range: 3. Sketch the graph of g() = cos and state the period and zeros of this function. g() Period: Zeros: (5) Module
Principles of Mathematics Section Assignment. 0 (3) 4. Graphicall solve the equation sin = cos where 0 < [Hint: Sketch two graphs and find their points of intersection.] () 5. If = p cos(q + r) + s, identif the meaning of each of the four constants p, q, r, and s. Describe in a sentence how the affect the graph of = cos. Module
0 Section Assignment. Principles of Mathematics. Sketch at least one period of the graph of = cos4. State the domain, range, period, and phase. () 7. Identif the equation of the graph below in the form = A sin B( C) + D (4) 3 Module
Principles of Mathematics Section Assignment. 03 (4) 8. Find the area of rectangle ABCD if the equation of the curve is = tan. A D 4 B C Total: 40 marks Send in this work as soon as ou complete this section. Module
04 Section Assignment. Principles of Mathematics Module