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

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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

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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)

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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)

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

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General Information you already know

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1: Graph Tangent

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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

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1: Graph Cotangent undefined

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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

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1: Graph Cosecant 0

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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

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1: Graph Secant = /2+ 2 n

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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.

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

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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.

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

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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.

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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

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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

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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

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Graphs of the Trig Functions Objective To use the graphs of the trigonometric functions.

Graphs of the Trig Functions Objective To use the graphs of the trigonometric functions.

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