Graphs of Secant, Cosecant, and Cotangent Section 4.6 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review The Unit Circle 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review Unit Circle of Sin Coordinates 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Quadrantals From the Unit Circle, there are five points to graph. Secant and Cosecant graph points are at 𝟎, 𝝅 𝟐 , 𝝅, 𝟑𝝅 𝟐 , and 𝟐𝝅 Cotangent graph points are at 𝟎, 𝝅 𝟒 , 𝝅 𝟐 , 𝟑𝝅 𝟒 ,𝝅 Cosecant graph is the reciprocal of a sine function, Secant graph is the reciprocal function of cosine function, and Cotangent graph is the reciprocal of the tangent function 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review 𝒙 𝒇(𝒙) 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝒇(𝒙) 𝟎 𝟏 −𝟏 Graph 𝒚= 𝐬𝐢𝐧 𝒙 from 𝟎,𝟐𝝅 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 1 Graph 𝒚=𝐜𝐬𝐜 𝒙 from 𝟎,𝟐𝝅 Radians 𝒇(𝒙) 𝟎 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟐𝝅 𝐬𝐢𝐧 𝒙 𝟎 𝟏 −𝟏 𝟏/𝐬𝐢𝐧 𝒙 𝑼𝒏𝒅 𝟏 −𝟏 𝟏/𝐬𝐢𝐧 𝒙 𝟏/𝟎 𝟏/𝟏 𝟏/−𝟏 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 1 𝒙 𝒇(𝒙) 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝐬𝐢𝐧 𝒙 𝟎 𝟏 −𝟏 𝟏/𝐬𝐢𝐧 𝒙 𝐔𝐧𝐝 𝟏 −𝟏 Graph 𝒚= 𝐜𝐬𝐜 𝒙 from 𝟎,𝟐𝝅 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 1 Graph 𝒚= 𝐜𝐬𝐜 𝒙 from 𝟎,𝟐𝝅 Period: Domain: Range: Odd or Even Function? 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 1 𝒙 𝐜𝐬𝐜(𝒙) 𝟎 𝐔𝐧𝐝 𝝅/𝟐 𝟏 𝝅 𝟑𝝅/𝟐 −𝟏 𝟐𝝅 Graph 𝒚= 𝐜𝐬𝐜 𝒙 from 𝟎,𝟐𝝅 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review Unit Circle of Cosine Coordinates 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review 𝒙 𝒇(𝒙) 𝟎 𝝅/𝟐 𝝅 𝟑𝝅/𝟐 𝟐𝝅 𝒇(𝒙) 𝟏 𝟎 −𝟏 Graph 𝒚= 𝐜𝐨𝐬 𝒙 from 𝟎,𝟐𝝅 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 2 Graph 𝒚 = 𝐬𝐞𝐜 𝒙 from 𝟎,𝟐𝝅 Radians 𝒇(𝒙) 𝟎 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟐𝝅 𝐜𝐨𝐬 𝒙 𝟏 𝟎 −𝟏 𝟏/𝐜𝐨𝐬 𝒙 𝟏 𝑼𝒏𝒅 −𝟏 𝟏/𝐬𝐢𝐧 𝒙 𝟏/𝟎 𝟏/𝟏 𝟏/−𝟏 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 2 Radians 𝒇(𝒙) 𝟎 𝝅 𝟐 𝝅 𝟑𝝅 𝟐 𝟐𝝅 𝐜𝐨𝐬 𝒙 𝟏 𝟎 −𝟏 𝟏/𝐜𝐨𝐬 𝒙 𝟏 𝑼𝒏𝒅 −𝟏 Graph 𝒚 = 𝐬𝐞𝐜 𝒙 from 𝟎,𝟐𝝅 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 2 Graph 𝒚= 𝐬𝐞𝐜 𝒙 from 𝟎,𝟐𝝅 Period: Domain: Range: Odd or Even Function? 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review Unit Circle of Tangent Coordinates 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Review Graph 𝒚 = 𝐭𝐚𝐧(𝒙) from − 𝝅 𝟐 , 𝝅 𝟐 Period: Domain: Range: Zeroes: Odd or Even Function? 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Radians 𝒇(𝒙) 𝟎 𝝅/𝟒 𝝅/𝟐 𝟑𝝅/𝟒 𝝅 𝒄𝒐𝒕⁡𝒙 𝐔𝐧𝐝 𝟏 𝟎 −𝟏 Example 3 Graph 𝒚=𝐜𝐨𝐭⁡(𝒙) from 𝟎,𝝅 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 3 Graph 𝒚=𝐜𝐨𝐭⁡(𝒙) from 𝟎,𝝅 Period: Domain: Range: Zeroes: Odd or Even Function? 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Transformations 𝒚= 𝑨 𝐭𝐫𝐢𝐠 𝐟𝐮𝐧𝐜𝐭𝐢𝐨𝐧 𝑩 𝒙−𝒄 +𝒅 𝑨 is the amplitude (only exist with sine and cosine graphs) 𝑩 is the period or frequency 𝒄 is the vertical shift 𝒅 is the horizontal shift or phase shift Period is over 𝟐𝝅 for all functions except Tan and Cot. Period is 𝝅 for Tan and Cot. 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 4 Identify the period and amplitude of 𝒚=𝟐 𝐜𝐬𝐜 𝟐𝒙 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Example 5 Identify the period and amplitude of 𝒚= 𝟏 𝟒 𝐜𝐨𝐭 𝟑(𝒙−𝝅)+𝟐 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Your Turn Identify the period and amplitude of 𝒚= 𝟏 𝟐 𝐜𝐬𝐜 𝟑𝒙+𝟏 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot

§4.6: Graphs of Sec, Csc, and Cot Assignment Worksheet 2/24/2019 7:54 PM §4.6: Graphs of Sec, Csc, and Cot