College Trigonometry

Transcription

1 College Trigonometry George Voutsadakis 1 1 Mathematics and Computer Science Lake Superior State University LSSU Math 11 George Voutsadakis (LSSU) Trigonometry January / 8

2 Outline 1 Trigonometric Functions Angles and Arcs Right Triangle Trigonometry Trigonometric Functions of Any Angle Trigonometric Functions of Real Numbers Graphs of the Sine and Cosine Functions Graphs of the Other Trigonometric Functions Graphing Techniques George Voutsadakis (LSSU) Trigonometry January 015 / 8

3 Angles and Arcs Subsection 1 Angles and Arcs George Voutsadakis (LSSU) Trigonometry January 015 / 8

4 Angles and Arcs Terminology on Angles The two parts into which a point P on a line separates the line are called half-lines or rays; The half-line formed by P that includes a point A on the line is denoted by PA; P is the endpoint of PA; Definition of Angle An angle is formed by rotating a given ray about its endpoint to some terminal position; The original ray is called the initial side of the angle and the second ray is the terminal side; The common endpoint is the vertex of the angle. Angles formed by a counterclockwise rotation are positive angles and those formed by a clockwise rotation are negative angles; Notation for angles: α = AOB; β = CPD; γ = FQE; George Voutsadakis (LSSU) Trigonometry January / 8

5 Angles and Arcs Degree Measure Degree Measure An angle formed by rotating the initial side counterclockwise exactly once until it coincides with itself is defined to have a measure of 60 degrees, written 60 ; Therefore, one degree is the measure of an angle formed by rotating a ray 1 60 of a complete revolution and it is written 1 ; Angles are classified according to their degree measure: George Voutsadakis (LSSU) Trigonometry January / 8

6 Angles and Arcs Standard Position, Complementary and Supplementary An angle superimposed in a Cartesian coordinate system is in standard position if its vertex is at the origin and its initial side is on the positive x-axis: Two angles are coterminal if they share the same terminal side when placed in standard position; Two positive angles are complementary if the sum of their measures is 90 and they are supplementary if the sum of their measures is 180 ; George Voutsadakis (LSSU) Trigonometry January / 8

7 Angles and Arcs Simple Examples Find, if possible the measure of the complement and the supplement of θ = 40 ; Comp(θ) = = 50 ; Supp(θ) = = 140 ; Find, if possible the measure of the complement and the supplement of θ = 15 ; θ does not have a complement since it is an angle with measure greater than 90 ; Supp(θ) = = 55 ; Are the two acute angles of any right triangle complementary angles? Yes! because their sum is 90 ; George Voutsadakis (LSSU) Trigonometry January / 8

8 Angles and Arcs Quadrantal Angles An angle is a quadrantal angle if its terminal side in standard position lies on a coordinate axis; For instance, the 90,180 and 70 angles are all quadrantal angles; Recall that two angles are coterminal if they share the same terminal side when placed in standard position; Measures of Coterminal Angles Given an angle θ in standard position with measure x, then the measures of the angles that are coterminal with θ are given by x +k 60, where k is an integer. George Voutsadakis (LSSU) Trigonometry January / 8

9 Angles and Arcs An Example Assume that the following angles are in standard position; Determine the measure of the positive angle with measure less than 60 that is coterminal with the given angle and classify the angle by quadrant; α = 550 ; We have α = 550 = ; Therefore, α is coterminal with the 190 angle and the terminal side lies in Quadrant III; β = 55 ; We have β = 55 = ; Therefore, β is coterminal with the 105 angle and the terminal side lies in Quadrant II; γ = 1105 ; We have γ = 1105 = ; Therefore, γ is coterminal with the 5 angle and the terminal side lies in Quadrant I; George Voutsadakis (LSSU) Trigonometry January / 8

10 Angles and Arcs Decimal Degrees and DMS (Degree, Minute, Second) To represent a fractional part of a degree, there are two popular methods: The decimal degree method uses a decimal number; For instance, 4.4 means 4 plus 4 hundredths of 1 ; The DMS (Degree, Minute, Second) method subdivides a degree into 60 minutes (1 = 60 ) and each minute into 60 seconds (1 = 60 ); Example: Write as a decimal degree; = ( ) = ( ) = ; George Voutsadakis (LSSU) Trigonometry January / 8

11 Angles and Arcs Central Angles and the Radian Consider a circle of radius r and two radii OA and OB; The angle θ formed by OA and OB is called a central angle; The portion of the arc between A and B is an arc of the circle and is denoted by AB ; The arc AB is said to subtend the angle θ; Definition of a Radian Oneradianisdefinedtobethemeasureofthecentral angle subtended by an arc of length r on a circle of radius r; George Voutsadakis (LSSU) Trigonometry January / 8

12 Angles and Arcs Radian Measure of an Angle Definition of Radian Measure Given an arc of length s on a circle of radius r, the measure of the central angle subtended by the arc is θ = s r radians. Example: Suppose an arc has length 15 cm on a circle of radius 5 cm. What is the radian measure of the central angle subtended by the arc? θ = s r = 15 5 = radians; Example: An arc of length 1 cm has radian measure 4 radians; What is the radius of the corresponding circle? θ = s r = s r = 1 r θ 4/ = 9 cm; George Voutsadakis (LSSU) Trigonometry January / 8

13 Angles and Arcs Conversion Between Radians and Degrees Radian-Degree Conversions To convert from radians to degrees, multiply by 180 π rads ; To convert from degrees to radians, multiply by π rads 180 ; Example: Convert from degrees to radians: 60 = 60 π rads 180 = π rads; 15 = 15 π rads 180 = 7π 4 rads; 150 = 150 π rads 180 = 5π 6 rads; Example: Convert from radians to degrees: π 4 rads = π 4 rads 180 π rads = 15 ; ; rad = 1 rad π rads = 180 π 5π rads = 5π rads 180 π rads = 450 ; George Voutsadakis (LSSU) Trigonometry January / 8

14 Angles and Arcs Arc and Arc Length The length s of the arc subtending a central angle of nonnegative radian measure θ of a circle of radius r is given by s = rθ; Example: What is the length of the arc that subtends a central angle of 10 in a circle of radius 10 cm? First, convert degrees to radians: 10 = 10 π rads 180 = π rads; Then, use the formula: s = rθ = 10 cm π rad = 0π cm; George Voutsadakis (LSSU) Trigonometry January / 8

15 Angles and Arcs A More Challenging Application A pulley with a radius of 10 inches uses a belt to drive a pulley with a radius of 4 inches; Find the angle through which the smaller pulley turns as the 10-inch pulley makes one full revolution; State answer in both radians and degrees; For the large pulley, through one revolution we obtain s 1 = r 1 θ 1 = 10 in π rads = 0π in; During that revolution, since the two pulleys are connected through the belt, we get s = s 1 ; Therefore, s 1 = r θ θ = s 1 = 0π = 5π rads; r In degrees 5π rads = 5π rads π rads = 900 ; George Voutsadakis (LSSU) Trigonometry January / 8

16 Angles and Arcs Linear and Angular Speed and Their Relation Definition of Linear and Angular Speed Suppose that a point moves on a circular path of radius r at a constant rate of θ radians per unit of time t; If s is the distance that the point travels, then s = rθ; The linear speed of the point is v = s ; The angular t speed of the point is ω = θ t ; To reveal the relation between the linear and the angular speeds, note that v = s t = rθ t = rθ t = rω; Example: A hard disk rotates at 700 revolutions per minute; What is its angular speed in radians per second? 700 rev/min = 700 rev π rad min 1 rev 1 min = 40π rad/sec; 60 sec George Voutsadakis (LSSU) Trigonometry January / 8

17 Angles and Arcs Another Example A windmill has blades that are 1 feet in length; If it is rotating at revolutions per second, what is the linear speed in feet per second of the tips of the blades; The angular speed of the point is: ω = rev sec πrad rev = 6πrad sec. Thus, the linear speed is v = rω = 1 ft 6π rad sec = 7π ft/sec; George Voutsadakis (LSSU) Trigonometry January / 8

18 Right Triangle Trigonometry Subsection Right Triangle Trigonometry George Voutsadakis (LSSU) Trigonometry January / 8

19 Right Triangle Trigonometry Definitions of Trigonometric Functions Consider an acute angle θ of a right triangle; We refer to the vertical side opposite and the vertical side adjacent to the angle θ; Definition of Trigonometric Functions of θ The values of the trigonometric functions of θ are defined as follows: sinθ = opp cosθ = adj hyp hyp tanθ = opp adj secθ = hyp adj cotθ = adj opp cscθ = hyp opp George Voutsadakis (LSSU) Trigonometry January / 8

20 Right Triangle Trigonometry Computing Trig Functions Find the values of the trig functions of the angle θ of the triangle given in the figure First, compute the length c of the hypothenuse using the Pythagorean Theorem: c = a +b = +4 = 5 c = 5; Now set up the trig functions of θ: sinθ = opp hyp = adj ; cosθ = 5 hyp = 4 opp ; tanθ = 5 adj = 4 ; cotθ = adj opp = 4 hyp ; secθ = adj = 5 hyp ; cscθ = 4 opp = 5 ; George Voutsadakis (LSSU) Trigonometry January / 8

21 Right Triangle Trigonometry A More Challenging Example Given that θ is an acute angle and cosθ = 5, compute tanθ; 8 Since cosθ = adj hyp = 5, we get the 8 following diagram: Now, compute the length a of the opposite side to θ using the Pythagorean Theorem: a = c b = 8 5 = 64 5 = 9; a = 9; Therefore, we obtain tanθ = opp 9 adj = 5 ; George Voutsadakis (LSSU) Trigonometry January / 8

22 Right Triangle Trigonometry Trigonometric Numbers of θ = 45 We compute the trigonometric numbers of a 45 angle; Since a right triangle having a 45 angle is isosceles, we get the following diagram: Therefore, for the trigonometric numbers, we get: sin45 = opp hyp = x x = ; cos45 = adj hyp = x x = ; tan45 = opp adj = x x = 1; cot45 = adj opp = x x = 1; sec45 = hyp x adj = = ; csc45 = hyp x x opp = = ; x George Voutsadakis (LSSU) Trigonometry January 015 / 8

23 Right Triangle Trigonometry Trigonometric Numbers of θ = 0 and θ = 60 We compute the trigonometric numbers of a 0 and of a 60 angle; Since a right triangle having a 60 angle is half of an equilateral triangle, we get the following diagram: Therefore, for the trigonometric numbers, we get: sin0 = cos60 = x x = 1 x ; cos0 = sin60 = x = ; tan0 = cot60 = x x = x ; cot0 = tan60 = = ; x sec0 = csc60 = x x = ; csc0 = sec60 = x x = ; George Voutsadakis (LSSU) Trigonometry January 015 / 8

24 Right Triangle Trigonometry Table of Trigonometric Numbers of 0,45 and 60 θ sinθ cosθ tanθ cscθ secθ cotθ 0 ; π ; π ; π 1 George Voutsadakis (LSSU) Trigonometry January / 8

25 Right Triangle Trigonometry Evaluating Expressions Find the exact value of the following expressions: ( ) 1 sin 45 +cos 60 = +( csc π 4 sec π cos π 6 = ) = = 4 ; = ; George Voutsadakis (LSSU) Trigonometry January / 8

26 Right Triangle Trigonometry Reciprocal Identities Recall that we have sinθ = opp hyp cosθ = adj hyp tanθ = opp adj cscθ = hyp opp secθ = hyp adj cotθ = adj opp These imply the following important reciprocal identities: sinθ = 1 cscθ cosθ = 1 secθ tanθ = 1 cotθ cscθ = 1 sinθ secθ = 1 cosθ cotθ = 1 tanθ George Voutsadakis (LSSU) Trigonometry January / 8

27 Right Triangle Trigonometry Application: Angle of Elevation From apoint115 feet from thebase of a tree, the angle of elevation to the top of the tree is 64. ; What is the height of the tree? tan64. = opp adj = h 115 h = 115 tan ft. George Voutsadakis (LSSU) Trigonometry January / 8

28 Right Triangle Trigonometry Application: Angle of Depression Suppose the direct distance of a fighter jet from the landing deck of an aircraft carrier is 10 miles and the angle of depression is ; Find the horizontal ground distance from the jet to the carrier; cos = adj hyp = x 10 x = 10 cos 8.87 miles; George Voutsadakis (LSSU) Trigonometry January / 8

29 Right Triangle Trigonometry Application: Angle of Elevation Revisited An observer notes that the angle of elevation from a point A to the top of the Eiffel tower is 70 ; From another point 10 feet further from the base of the tower, the angle of elevation is 60 ; Find the height of the Eiffel tower; tan70 = h x = x Moreover, h tan70 = hcot70 ; tan60 h = x +10 = h hcot h = (tan60 )(hcot70 +10) h = htan60 cot70 +10tan60 h htan60 cot70 = 10tan60 10tan60 h = 1 tan feet; cot70 George Voutsadakis (LSSU) Trigonometry January / 8

30 Trigonometric Functions of Any Angle Subsection Trigonometric Functions of Any Angle George Voutsadakis (LSSU) Trigonometry January / 8

31 Trigonometric Functions of Any Angle Trigonometric Functions of Any Angle Trigonometric Functions of Any Angle Suppose P(x,y) is a point different from the origin on the terminal side of an angle θ in standard position, such that r = x +y is the distance from the origin to P; The six trigonometric functions of θ are defined as follows: sinθ = y r ; cosθ = x r ; tanθ = y,x 0; x cscθ = r y,y 0; secθ = r x,x 0; cotθ = x,y 0; y George Voutsadakis (LSSU) Trigonometry January / 8

32 Trigonometric Functions of Any Angle Evaluating Trigonometric Functions I Find the exact value of the six trigonometric functions of the angle θ in standard position whose terminal side contains the point P(, ); We get x =,y = and r = ( ) +( ) = 1; Thus, sinθ = y r = = ; cosθ = x r = = ; tanθ = y x = = ; secθ = cscθ = 1 ; cotθ = ; 1 ; George Voutsadakis (LSSU) Trigonometry January 015 / 8

33 Trigonometric Functions of Any Angle Quadrantal Angles and Signs of Functions Values of Trigonometric Functions of Quadrantal Angles: θ sinθ cosθ tanθ cscθ secθ cotθ Signs of Trigonometric Functions: Sign of Quadrant I Quadrant II Quadrant III Quadrant IV sinθ and cscθ + + cosθ and secθ + + tanθ and cotθ + + George Voutsadakis (LSSU) Trigonometry January 015 / 8

34 Trigonometric Functions of Any Angle Evaluating Trigonometric Functions II Given tanθ = 7 5 and sinθ < 0, find cosθ and cscθ; Since tanθ = 7 5 and sinθ < 0, we get y x = 7 5 Therefore y = 7 and x = 5; These imply that r = 5 +( 7) = 74; Therefore cosθ = x r = 5 = and cscθ = r y = 74 7 ; and y < 0; George Voutsadakis (LSSU) Trigonometry January / 8

35 Trigonometric Functions of Any Angle The Reference Angle Given θ in standard position, its reference angle θ is the acute angle formed by the terminal side of θ and the x-axis; Example: Find the measure of the reference angle θ for each of the following: θ = 10 ; Since 10 = , we have θ = 60 ; θ = 45 ; Since 45 = 60 15, we have θ = 15 ; θ = 7π 4 ; Since 7π 4 = π π 4, we have θ = π 4 ; θ = 1π 6 ; Since 1π 6 = π+ π 6, we have θ = π 6 ; George Voutsadakis (LSSU) Trigonometry January / 8

36 Trigonometric Functions of Any Angle Using the Reference Angle Reference Angle Theorem To evaluate sinθ, determine sinθ ; Then use either sinθ or sinθ, depending on which of the two has the correct sign. Example: Determine the exact value of sin10 ; We have θ = 0 and 10 is in Quadrant III; Thus, sin10 = sin0 = 1 ; cos405 ; We have θ = 45 and 405 is in Quadrant I; Thus, cos405 = cos45 = ; tan 5π ; We have θ = π and 5π is in Quadrant IV; Thus, tan 5π = tan π = ; George Voutsadakis (LSSU) Trigonometry January / 8

37 Trigonometric Functions of Real Numbers Subsection 4 Trigonometric Functions of Real Numbers George Voutsadakis (LSSU) Trigonometry January / 8

38 Trigonometric Functions of Real Numbers Wrapping Function Consider the unit circle, i.e., the circle of radius 1 centered at the origin; The wrapping function has domain all real numbers and maps a real number t to a point W(t) = P(x,y) on the unit circle such that the length of the arc AP is t, where A(1, 0); Since r = 1, we have s = 1 θ, i.e., the length s of the arc equals the measure θ of the central angle subtended by the arc! This allows one to associate an angle with any given real number t using the wrapping function and passing through the arc on the unit circle starting from A and having length t (clockwise if t < 0 and counterclockwise if t > 0); George Voutsadakis (LSSU) Trigonometry January / 8

39 Trigonometric Functions of Real Numbers An Example of Evaluating the Wrapping Function Evaluate W( π ); We have cos π = x r cos π = x 1 x = 1 ; Similarly, sin π = y r sin π = y 1 y = ; Therefore, the point W( π ) = ( 1, ); George Voutsadakis (LSSU) Trigonometry January / 8

40 Trigonometric Functions of Real Numbers Trigonometric Functions of Any Real Number Use the wrapping function on the unit circle to map a real number t, first to the arc AP of length t (counterclockwise for t > 0 and clockwise for t < 0) then to the central angle of measure t subtended by the arc AP ; We define the trigonometric functions of the real number t as the trigonometric functions of the angle corresponding to t, which (because r = 1) has measure t radians; Trigonometric Functions of Real Numbers Let t be a real number and W(t) = P(x,y); Then, we define sint = y, cost = x, tant = y,x 0, x csct = 1 y,y 0, sect = 1 x,x 0, cott = x,y 0; y George Voutsadakis (LSSU) Trigonometry January / 8

41 Trigonometric Functions of Real Numbers Evaluating Trigonometric Functions Find the exact value of each function: cos π ( 4 = ; sin 7π ) ( π = sin = 6 6) 1 ; ( tan 5π ) ( π = tan = 1; ( 4) 4) 5π ( π sec = sec = ; ) George Voutsadakis (LSSU) Trigonometry January / 8

42 Trigonometric Functions of Real Numbers Application: The Millenium Ferris Wheel in London The Millenium Wheel has a diameter of 450 feet and completes one revolution every 0 minutes; Suppose that the height h in feet above the Thames River of a person riding on the Wheel can be estimated by ( π ) h(t) = 55 5cos 15 t, where t in minutes is time since person started the ride; How high is the person at the start of the ride? h(0) = 55 5cos0 = = 0 feet; How high is the person after 18 minutes? ( ) 18π h(18) = 55 5cos = 55 5cos 15 ( ) 6π 47 feet; 5 George Voutsadakis (LSSU) Trigonometry January / 8

43 Trigonometric Functions of Real Numbers Domains and Ranges of Trigonometric Functions The domain and ranges of the trigonometric functions: Function Domain Range y = sint R {y : 1 y 1} y = cost R {y : 1 y 1} y = tant {t : t (k +1)π } R y = csct {t : t kπ} {y : y 1 or y 1} y = sect {t : t (k +1)π } {y : y 1 or y 1} y = cott {t : t kπ} R George Voutsadakis (LSSU) Trigonometry January / 8

44 Trigonometric Functions of Real Numbers Even and Odd Trigonometric Functions The four trigonometric functions y = sint, y = csct, y = tant, y = cott are all odd functions; The two trigonometric functions y = cost, y = sect are both even functions; These statements imply the following even-odd identities: sin( t) = sint cos( t) = cost tan( t) = tant csc( t) = csct sec( t) = sect cot( t) = cott Example: Is f(x) = x tanx even, odd or neither? f( x) = ( x) tan( x) = x ( tanx) = x +tanx = (x tanx) = f(x); Therefore, f(x) is an odd function; George Voutsadakis (LSSU) Trigonometry January / 8

45 Trigonometric Functions of Real Numbers Periodicity A function f is periodic if there exists a positive constant p, such that f(t +p) = f(t) for all t in the domain of f; The smallest such positive p for which f is periodic is called the period of f; The functions y = sint, y = cost, y = csct, y = sect are periodic with period π; The functions y = tant, y = cott are periodic with period π; These statements imply the following periodic identities: sin(t +kπ) = sint cos(t +kπ) = cost tan(t +kπ) = tant csc(t +kπ) = csct sec(t +kπ) = sect cot(t +kπ) = cott George Voutsadakis (LSSU) Trigonometry January / 8

46 Trigonometric Functions of Real Numbers Trigonometric Identities Reciprocal Identities sint = 1 csct, cost = 1 csct, tant = 1 cott ; Ratio Identities tant = sint cost ; cost cott = sint ; Pythagorean Identities cos t +sin t = 1, 1+tan t = sec t, 1+cot t = csc t; George Voutsadakis (LSSU) Trigonometry January / 8

47 Trigonometric Functions of Real Numbers Example I Use the unit circle and the definitions of trigonometric functions to show that sin(t +π) = sint; If W(t) = (x,y), then W(t +π) = ( x, y); Therefore sin(t +π) = y = sint; George Voutsadakis (LSSU) Trigonometry January / 8

48 Trigonometric Functions of Real Numbers Example II Write the expression 1 sin t + 1 cos t 1 sin t + 1 cos t as a single term; cos t = sin tcos t + sin t sin tcos t = cos t +sin t sin tcos t 1 = sin tcos t 1 = sin t 1 cos t = csc tsec t; George Voutsadakis (LSSU) Trigonometry January / 8

49 Trigonometric Functions of Real Numbers Example III For π < t < π, write tant in terms of only sint; Therefore, we get cos t +sin t = 1 cos t = 1 sin t cost = ± 1 sin t π <t<π cost = 1 sin t; tant = sint cost = sint 1 sin t ; George Voutsadakis (LSSU) Trigonometry January / 8

50 Graphs of the Sine and Cosine Functions Subsection 5 Graphs of the Sine and Cosine Functions George Voutsadakis (LSSU) Trigonometry January / 8

51 Graphs of the Sine and Cosine Functions Graph of y = sinx We create a small table of values: π x y = sinx 0 π π 1 π 5π 6 π 1 0 x 7π 6 y = sinx 1 4π We plot the points and connect: π 1 5π 11π 6 π 1 0 George Voutsadakis (LSSU) Trigonometry January / 8

52 Graphs of the Sine and Cosine Functions Basic Properties of y = sinx Extending the previous graph by periodicity, we get This graph has the following basic properties: Domain: All real numbers; Range: {y : 1 y 1}; Period: π; Symmetry: With respect to the origin (Odd); x-intercepts: kπ; George Voutsadakis (LSSU) Trigonometry January / 8

53 Graphs of the Sine and Cosine Functions Graph of y = asinx The amplitude of a graph with maximum value y = M and minimum value y = m is defined by A = 1 (M m); Example: y = sinx has M = 1 and m = 1; Thus, it has amplitude A = 1 (1 ( 1)) = 1; Amplitude of y = asinx The amplitude of y = asinx is a. Example: Graph y = sinx; To graph this function, we start from y = sinx, obtain y = sinx by a vertical stretch by a factor of and then obtain y = sinx by flipping with respect to the x-axis; George Voutsadakis (LSSU) Trigonometry January / 8

54 Graphs of the Sine and Cosine Functions Graph of y = sinbx Period of y = sinbx The period of y = sinbx is π b. Example: Find the amplitude and periods of the following functions: Function y = asinbx y = sin( x) y = sin x y = sin x 4 Amplitude a = 1 = 1 = π π Period b = π π 1/ = 6π π /4 = 8π Example: Graph y = sinπx; To graph this function, we start from y = sinx, obtain y = sinπx by a horizontal compression by a factor of π and then obtain y = sinπx by a vertical stretch by a factor of ; George Voutsadakis (LSSU) Trigonometry January / 8

55 Graphs of the Sine and Cosine Functions Graph of y = asinbx Example: Graph y = 1 sin x ; To graph this function, we start from y = sinx, obtain y = sin x by a horizontal stretching by a factor of, then obtain y = 1 sin x by a vertical compression by a factor of and, finally, obtain y = 1 sin x by a flipping with respect to the x-axis; George Voutsadakis (LSSU) Trigonometry January / 8

56 Graphs of the Sine and Cosine Functions Graph of y = cosx We create a small table of values: x 0 π y = cosx 1 6 π π π π 6 π 1 x y = cosx 7π 6 We plot the points and connect: 4π π 5π π 6 π 1 George Voutsadakis (LSSU) Trigonometry January / 8

57 Graphs of the Sine and Cosine Functions Basic Properties of y = cosx Extending the previous graph by periodicity, we get This graph has the following basic properties: Domain: All real numbers; Range: {y : 1 y 1}; Period: π; Symmetry: With respect to the y-axis (Even); x-intercepts: (k +1) π ; George Voutsadakis (LSSU) Trigonometry January / 8

58 Graphs of the Sine and Cosine Functions Graph of y = acosx Recall that the amplitude of a graph is defined by A = 1 (M m); Example: y = cosx has M = 1 and m = 1; Thus, it has amplitude A = 1 (1 ( 1)) = 1; Amplitude of y = acosx The amplitude of y = acosx is a. Example: Graph y = 5 sinx; To graph this function, we start from y = cosx, obtain y = 5 cosx by a vertical stretch by a factor of 5 and then obtain y = 5 cosx by flipping with respect to the x-axis; George Voutsadakis (LSSU) Trigonometry January / 8

59 Graphs of the Sine and Cosine Functions Graph of y = cosbx Period of y = cosbx The period of y = cosbx is π b. Example: Find the amplitude and periods of the following functions: Function y = acosbx y = cosx y = cos x Amplitude a = = π π π Period b / = π Example: Graph y = cos π x; To graph this function, we start from y = cosx, obtain y = cos π x by a horizontal compression by a factor of π and then obtain y = cos π x by a vertical stretch by a factor of ; George Voutsadakis (LSSU) Trigonometry January / 8

60 Graphs of the Sine and Cosine Functions Graph of y = acosbx Example: Graph y = cos πx 4 ; To graph this function, we start from y = cosx, obtain y = cos πx 4 by a horizontal stretching by a factor of 4 πx, then obtain y = cos π 4 by a vertical stretching by a factor of and, finally, obtain y = cos πx 4 by a flipping with respect to the x-axis; George Voutsadakis (LSSU) Trigonometry January / 8

61 Graphs of the Sine and Cosine Functions Finding an Equation for a Graph I The graph on the right shows a single cycle of a graph of a sine or cosine function; Find an equation for the graph; The graph has Amplitude a = ; Period T = 6 π b = 6 b = π 6 = π b = ±π ; Since at x = 0, it has value y = +, we get an equation y = acosbx = cos π x; George Voutsadakis (LSSU) Trigonometry January / 8

62 Graphs of the Sine and Cosine Functions Finding an Equation for a Graph II The graph on the right shows a single cycle of a graph of a sine or cosine function; Find an equation for the graph; The graph has Amplitude a = ; Period T = 4π π b = 4π π b = 4π/ = b = ± ; Since at x = π, it has value y =, we get an equation y = asinbx = sin x; George Voutsadakis (LSSU) Trigonometry January / 8

63 Graphs of the Other Trigonometric Functions Subsection 6 Graphs of the Other Trigonometric Functions George Voutsadakis (LSSU) Trigonometry January / 8

64 Graphs of the Other Trigonometric Functions Graph of y = tanx We create a small table of values: x 0 π y = tanx 0 π π π We plot the points, connect and use the odd property: George Voutsadakis (LSSU) Trigonometry January / 8

65 Graphs of the Other Trigonometric Functions Basic Properties of y = tanx Extending the previous graph by periodicity, we get This graph has the following basic properties: Domain: R {(k +1) π : k Z}; Range: All reals; Period: π; Symmetry: With respect to the origin (Odd); x-intercepts: kπ; Vertical Asymptotes: x = (k +1) π ; George Voutsadakis (LSSU) Trigonometry January / 8

66 Graphs of the Other Trigonometric Functions Graph of y = atanx The graph of y = atanx does not have an amplitude since it does not have a maximum or minimum value; It just represents either a vertical stretching or a vertical compression of the graph of y = ±tanx; Example: Graph y = 1 5 tanx; To graph this function, we start from y = tanx, obtain y = 1 tanx by a 5 vertical compression by a factor of 5 and then obtain y = 1 tanx by 5 flipping with respect to the x-axis; George Voutsadakis (LSSU) Trigonometry January / 8

67 Graphs of the Other Trigonometric Functions Graph of y = tanbx Period of y = tanbx The period of y = tanbx is π b. Example: Graph y = tanπx; To graph this function, we start from y = tanx, obtain y = tanπx by a horizontal compression by a factor of π and then obtain y = tanπx by a vertical stretch by a factor of ; George Voutsadakis (LSSU) Trigonometry January / 8

68 Graphs of the Other Trigonometric Functions Graph of y = atanbx Example: Graph y = 1 tan x ; To graph this function, we start from y = tanx, obtain y = tan x by a horizontal stretching by a factor of, then obtain y = 1 tan x by a vertical compression by a factor of and, finally, obtain y = 1 tan x byaflippingwithrespect to the x-axis; George Voutsadakis (LSSU) Trigonometry January / 8

69 Graphs of the Other Trigonometric Functions Graph of y = cotx We create a small table of values: x 0 y = cotx 1 0 We plot the points, connect and use the odd property: π 6 π 4 π π George Voutsadakis (LSSU) Trigonometry January / 8

70 Graphs of the Other Trigonometric Functions Basic Properties of y = cotx Extending the previous graph by periodicity, we get This graph has the following basic properties: Domain: R {kπ : k Z}; Range: All reals; Period: π; Symmetry: With respect to the origin (Odd); x-intercepts: (k +1) π ; Vertical Asymptotes: x = kπ; George Voutsadakis (LSSU) Trigonometry January / 8

71 Graphs of the Other Trigonometric Functions Graph of y = acotbx Period of y = acotbx The period of y = acotbx is π b. Example: Graph y = cot x ; To graph this function, we start from y = cotx, obtain y = cot x by a horizontal stretching by a factor of and then obtain y = cot x by a vertical stretch by a factor of ; George Voutsadakis (LSSU) Trigonometry January / 8

72 Graphs of the Other Trigonometric Functions Graph of y = cscx We create a small table of values: x 0 π 6 y = cscx π 4 π π 1 π π 4 We plot the points, connect and use the odd property: 5π 6 π George Voutsadakis (LSSU) Trigonometry January / 8

73 Graphs of the Other Trigonometric Functions Basic Properties of y = cscx Extending the previous graph by periodicity, we get This graph has the following basic properties: Domain: R {kπ : k Z}; Range: {y : y 1 or y 1}; Period: π; Symmetry: With respect to the origin (Odd); x-intercepts: None; Vertical Asymptotes: x = kπ; George Voutsadakis (LSSU) Trigonometry January / 8

74 Graphs of the Other Trigonometric Functions Graph of y = acscbx Period of y = acscbx The period of y = acscbx is π b. Example: Graph y = 1 πx csc ; To graph this function, we start from y = cscx, obtain y = csc πx by a horizontal compression by a factor of π and then obtain y = 1 πx csc by a vertical compression by a factor of ; George Voutsadakis (LSSU) Trigonometry January / 8

75 Graphs of the Other Trigonometric Functions Graph of y = secx We create a small table of values: x 0 y = secx 1 π 6 π 4 π We plot the points and connect: π π π 4 5π 6 π 1 George Voutsadakis (LSSU) Trigonometry January / 8

76 Graphs of the Other Trigonometric Functions Basic Properties of y = secx Extending the previous graph by symmetry and periodicity, we get This graph has the following basic properties: Domain: R {(k +1) π : k Z}; Range: {y : y 1 or y 1}; Period: π; Symmetry: With respect to the x-axis (Even); x-intercepts: None; Vertical Asymptotes: x = (k +1) π ; George Voutsadakis (LSSU) Trigonometry January / 8

77 Graphs of the Other Trigonometric Functions Graph of y = asecbx Period of y = asecbx The period of y = asecbx is π b. Example: Graph y = sec x ; To graph this function, we start from y = secx, obtain y = sec x by a horizontal stretching by a factor of, then obtain y = sec x by a vertical stretching by a factor of and, finally, y = sec x by flipping with respect to the x-axis; George Voutsadakis (LSSU) Trigonometry January / 8

78 Graphing Techniques Subsection 7 Graphing Techniques George Voutsadakis (LSSU) Trigonometry January / 8

79 Graphing Techniques Amplitude, Period and Phase Shift of Sinusoidal Graphs Graphs of y = asin(bx +c) and y = acos(bx +c) The graphs of y = asin(bx +c) and y = acos(bx +c) have Amplitude : a, Period : π b, Phase Shift : c b ; The graph y = asin(bx +c) shifts the graph of y = asinbx horizontally c b units; The graph y = acos(bx +c) shifts the graph of y = acosbx horizontally c b units; Example: What is the phase shift of y = sin( 1 x π 6 )? φ = c b = π/6 1/ = π ; George Voutsadakis (LSSU) Trigonometry January / 8

80 Graphing Techniques Using Amplitudes, Periods and Phase Shifts to Graph Find the amplitude, period and phase shift of y = cos(x + π ) and use them to sketch the graph; Amplitude: a = ; Period:T = π b = π = π; Phase Shift:φ = c b = π/ = π 6 ; George Voutsadakis (LSSU) Trigonometry January / 8

81 Graphing Techniques Period and Phase Shift of Tangent and Cotangent Graphs of y = atan(bx +c) and y = acot(bx +c) The graphs of y = atan(bx +c) and y = acot(bx +c) have π Period : b, Phase Shift : c b ; The graph y = atan(bx +c) shifts the graph of y = atanbx horizontally c b units; The graph y = acot(bx +c) shifts the graph of y = acotbx horizontally c b units; Example: graph one period of y = cot(x ); Since the period is T = π b = π and the phase shift is φ = c b = =, we start the graph at x = and end it at x = + π = π+ ; George Voutsadakis (LSSU) Trigonometry January / 8

82 Graphing Techniques Using Amplitudes, Periods and Shifts to Graph I Graph y = 1 sin(x π 4 ) ; Amplitude: a = 1 ; π Period:T = b = π 1 = π; Phase Shift:φ = c b = π/4 = π 1 4 ; Vertical Shift:y 0 = ; George Voutsadakis (LSSU) Trigonometry January / 8

83 Graphing Techniques Using Amplitudes, Periods and Shifts to Graph II Find the amplitude, period and phase shift of y = cos(πx + π )+1 and used them to sketch the graph; Amplitude: a = ; Period:T = π b = π π = ; Phase Shift:φ = c b = π/ π = 1 ; Vertical Shift:y 0 = 1; George Voutsadakis (LSSU) Trigonometry January / 8