1. Graphs of Secant, Cosecant, and Cotangent
Section 4.6
Precalculus PreAP/Dual, Revised ©2017
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§4.6: Graphs of Sec, Csc, and Cot

2. §4.6: Graphs of Sec, Csc, and Cot
Review
The Unit Circle
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§4.6: Graphs of Sec, Csc, and Cot

3. §4.6: Graphs of Sec, Csc, and Cot
Review
Unit Circle of Sin Coordinates
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§4.6: Graphs of Sec, Csc, and Cot

4. §4.6: Graphs of Sec, Csc, and Cot
Review
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§4.6: Graphs of Sec, Csc, and Cot

5. §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
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§4.6: Graphs of Sec, Csc, and Cot

6. §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
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§4.6: Graphs of Sec, Csc, and Cot

7. §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
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§4.6: Graphs of Sec, Csc, and Cot

8. §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
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§4.6: Graphs of Sec, Csc, and Cot

9. §4.6: Graphs of Sec, Csc, and Cot
Example 1
Graph 𝒚= 𝐜𝐬𝐜 𝒙 from 𝟎,𝟐𝝅
Period:
Domain:
Range:
Odd or Even Function?
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§4.6: Graphs of Sec, Csc, and Cot

10. §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
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§4.6: Graphs of Sec, Csc, and Cot

11. §4.6: Graphs of Sec, Csc, and Cot
Review
Unit Circle of Cosine Coordinates
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§4.6: Graphs of Sec, Csc, and Cot

12. §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
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§4.6: Graphs of Sec, Csc, and Cot

13. §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
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§4.6: Graphs of Sec, Csc, and Cot

14. §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
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§4.6: Graphs of Sec, Csc, and Cot

15. §4.6: Graphs of Sec, Csc, and Cot
Example 2
Graph 𝒚= 𝐬𝐞𝐜 𝒙 from 𝟎,𝟐𝝅
Period:
Domain:
Range:
Odd or Even Function?
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§4.6: Graphs of Sec, Csc, and Cot

16. §4.6: Graphs of Sec, Csc, and Cot
Review
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§4.6: Graphs of Sec, Csc, and Cot

17. §4.6: Graphs of Sec, Csc, and Cot
Review
Unit Circle of Tangent Coordinates
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§4.6: Graphs of Sec, Csc, and Cot

18. §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?
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§4.6: Graphs of Sec, Csc, and Cot

19. §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
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§4.6: Graphs of Sec, Csc, and Cot

20. §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?
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§4.6: Graphs of Sec, Csc, and Cot

21. §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.
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§4.6: Graphs of Sec, Csc, and Cot

22. §4.6: Graphs of Sec, Csc, and Cot
Example 4
Identify the period and amplitude of 𝒚=𝟐 𝐜𝐬𝐜 𝟐𝒙
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§4.6: Graphs of Sec, Csc, and Cot

23. §4.6: Graphs of Sec, Csc, and Cot
Example 5
Identify the period and amplitude of 𝒚= 𝟏 𝟒 𝐜𝐨𝐭 𝟑(𝒙−𝝅)+𝟐
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§4.6: Graphs of Sec, Csc, and Cot

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

25. §4.6: Graphs of Sec, Csc, and Cot
Assignment
Worksheet
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§4.6: Graphs of Sec, Csc, and Cot