TRIGONOMETRY, 5.0 STUDENTS KNOW THE DEFINITIONS OF THE TANGENT AND COTANGENT FUNCTIONS AND CAN GRAPH THEM. Graphing Other Trigonometric Functions

Objective Key Words 1. Graph tangent, cotangent, secant, and cosecant functions. 2. Write equations of trigonometric functions Tangent Cotangent Secant Cosecant Domain Range X-intercept Y-intercept Asymptote Graphing Other Trigonometric Functions

Quick Check How many completely whole apples do you have if you have 5/4 of an apple? So what is left? How many completely whole apples do you have if you have ½ of an apple? So what is left? How many completely whole apples do you have if you have 8 apples? So what is left? How would you express these three questions as an algebraic expression? (Hint: apples, pieces of apples)

Quick Check Now think of π as the apple.  How many completely whole π do you have if you have 5/4 of an π? So what is left?  How many completely whole π do you have if you have ½ of an π? So what is left?  How many completely whole π do you have if you have 8 π? So what is left?  How would you express these three questions as an algebraic expression? (Hint: π, pieces of π known as remainder)

Trigonometric functions Reciprocal of Trigonometric functions Before We Begin, Recall the Unit Circle:

General Information you already know

1: Graph Tangent

Example for Tangent of an Angle Find each value by referring to the graphs of the trigonometric functions. tan 11π/4 Since 11π/4 = 2  + 3π/4, Then tan 11π/4 = -1. You try: tan 7  /2

1: Graph Cotangent undefined

Example for Cotangent of an Angle Find each value by referring to the graphs of the trigonometric functions. cot 11π/4 Since 5π/4 = 2  + π/2, Then cot 5π/4 = 0. You try: cot 3  /2

1: Graph Cosecant 0

Example for Cosecant of an Angle Find the values of  for which each equation is true. csc  = -1 From the pattern of the cosecant function, csc  =-1 if  = 3  /2+ 2  n, where n is an integer. You try: csc θ = 1

1: Graph Secant  =  /2+ 2  n

Example for Cosecant of an Angle Find the values of  for which each equation is true. sec  = -1 From the pattern of the secant function, sec  = -1 if  =  n, where n is an odd integer. You try: sec θ = 1 From the pattern of the secant function, sec  = 1 if  =  n, where n is an even integer.

Order does matter! y=A ???[B(θ-h)]+k 2: Graphing Trigonometric Functions

2: Example for Graphing Graph y=csc(  -  /2)+1. The vertical shift is 1. Use this information to graph the function. Amplitude is 1. The period is 2  /1 or 2 . The phase shift -(-  /2/1) or  /2.

2: Example for Graphing YOU TRY! Graph y=csc(2  -  /2)+1.

2: Example for Graphing YOU TRY! Graph y=csc(2  -  /2)+1. The vertical shift is 1. Use this information to graph the function. The amplitude is 1 The period is 2  /2 or . The phase shift -(-  /2/2) or  /4.

2: Example for Graphing Write an equation for a secant function with period , phase shift –π/2, and vertical shift 3. Substitute these values into the general equation. The equation is y = sec (2  +  ) + 3. The vertical shift is k=3. Thus, midline y=3 The amplitude is 1. Thus, draw the dashed lines above and below the midline The period π. Thus, B=2. Draw the Secant curve The phase shift is h=-π/2

2: Example for Graphing YOU TRY. Write an equation for a secant function with period , phase shift π/3, and vertical shift -3. Substitute these values into the general equation. The equation is y = sec (2  -2  /3)-3. The vertical shift is k=-3. Thus, midline y=-3 The amplitude is 1. Thus, draw the dashed lines above and below the midline The period π. Thus, B=2. Draw the Secant curve The phase shift is h=π/3

Summary Assignment Remember the functions tangent and cotangent have a period of . Whereas sine and its reciprocal function cosecant and cosine and its reciprocal function secant both have periods of 2 . 6.7: Graphing Other Trigonometric Functions  Pg400#(13-43 ALL, 45,48 EC) Problems not finished are left as homework. Conclusions