1. Page 452 15) Sin: –0.9511; Cos: –0.3090, Tan: 3.0777 Quadrant III 17)1)
sin t: 𝟕 𝟓𝟑 𝟓𝟑 , cos t: 𝟐 𝟓𝟑 𝟓𝟑 , tan t: 𝟕 𝟐
csc t: 𝟓𝟑 𝟕 , sec t: 𝟓𝟑 𝟐 , cot t: 𝟐 𝟕 3)
sin t: − 𝟔 𝟔𝟏 𝟔𝟏 , cos t: − 𝟓 𝟔𝟏 𝟔𝟏 , tan t: 𝟔 𝟓
csc t: − 𝟔𝟏 𝟔 , sec t: − 𝟔𝟏 𝟓 , cot t: 𝟓 𝟔 5)
sin t: − 𝟏𝟎 𝟏𝟎𝟑 𝟏𝟎𝟑 , cos t: − 𝟑𝟎𝟗 𝟏𝟎𝟑 , tan t: − 𝟏𝟎 𝟑 𝟑
csc t: − 𝟏𝟎𝟑 𝟏𝟎 , sec t: − 𝟑𝟎𝟗 𝟑 , cot t: − 𝟑 𝟏𝟎 7)
sin t: 𝟓 𝟓 , cos t: − 𝟐 𝟓 𝟓 , tan t: − 𝟏 𝟐
csc t: − 𝟓 , sec t: − 𝟓 𝟐 , cot t: −𝟐 9)
sin t: − 𝟒 𝟓 , cos t: − 𝟑 𝟓 , tan t: 𝟒 𝟑
csc t: − 𝟓 𝟒 , sec t: − 𝟓 𝟑 , cot t: 𝟑 𝟒
15)
Sin: –0.9511; Cos: –0.3090, Tan:
Quadrant III
17)
Sin: ; Cos: , Tan:
Quadrant I
19)
Sin: –0.8660; Cos: –0.5, Tan:
21)
Sin: –1; Cos: 0, Tan: undefined
Y-axis
23)
Sin: ; Cos: –0.2752, Tan: –3.4939
Quadrant II
4/23/2019 1:22 AM
6.4B - The Unit Circle

2. 4/23/2019 1:22 AM
6.4B - The Unit Circle

3. Pre-Calculus PreAP/Dual, Revised ©2014
The Unit Circle
Section 6.4B
Pre-Calculus PreAP/Dual, Revised ©2014
4/23/2019 1:22 AM
6.4B - The Unit Circle

4. Review sin Ѳ = csc Ѳ = cos Ѳ = sec Ѳ = tan Ѳ = cot Ѳ =
4/23/2019 1:22 AM
6.4B - The Unit Circle

5. Reciprocal Identities Theorem
sin θ= csc θ=
cos θ= sec θ=
tan θ= cot θ=
4/23/2019 1:22 AM
6.4B - The Unit Circle

6. Right Triangles – 45°, 45°, 90°
triangles (Drawing not to scale) Radius = 1
4/23/2019 1:22 AM
6.4B - The Unit Circle

7. Right Triangles – 30°, 60°, 90°
triangles (Drawing not to scale) Radius = 1
4/23/2019 1:22 AM
6.4B - The Unit Circle

8. Reminder
4/23/2019 1:22 AM
6.4B - The Unit Circle

9. The Unit Circle Activity
4/23/2019 1:22 AM
6.4B - The Unit Circle

10. The Unit Circle
4/23/2019 1:22 AM
6.4B - The Unit Circle

11. The Unit Circle
4/23/2019 1:22 AM
6.4B - The Unit Circle

12. The Unit Circle: π/4 Family
cos Ѳ
sin Ѳ
Reference Angle: 45°
π/4
45°
3π/4
135°
5π/4
225°
7π/4
315°
Reference Angle: 45°
π/4
45°
√2/2
3π/4
135°
–√2/2
5π/4
225°
7π/4
315°
4/23/2019 1:22 AM
6.4B - The Unit Circle

13. The Unit Circle: π/6 Family
cos Ѳ
sin Ѳ
Reference Angle: 30°
π/6
30°
5π/6
150°
7π/6
210°
11π/6
330°
Reference Angle: 30°
π/6
30°
√3/2
1/2
5π/6
150°
–√3/2
7π/6
210°
–1/2
11π/6
330°
4/23/2019 1:22 AM
6.4B - The Unit Circle

14. The Unit Circle: π/3 Family
cos Ѳ
sin Ѳ
Reference Angle: 60°
π/3
60°
2π/3
120°
4π/3
240°
5π/3
300°
Reference Angle: 60°
π/3
60°
1/2
√3/2
2π/3
120°
–1/2
4π/3
240°
–√3/2
5π/3
300°
4/23/2019 1:22 AM
6.4B - The Unit Circle

15. Example 1 Solve cos 45° without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

16. Example 2 Solve sin π/4 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

17. Example 3 Solve cos 7π/4 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

18. Example 4 Solve tan 7π/4 without a calculator 4/23/2019 1:22 AM
11-8: The Unit Circle

19. Your Turn Solve tan 9π/4 without a calculator 4/23/2019 1:22 AM
11-8: The Unit Circle

20. Example 5 Solve sin π/6 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

21. Example 6 Solve tan 11π/6 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

22. Example 7 Solve tan –π/3 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

23. Your Turn Solve tan 2π/3 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

24. Example 8 Solve sec 7π/4 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

25. Example 9 Solve csc 4π/3 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

26. Example 10 Solve cot 11π/6 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

27. Your Turn Solve csc –11π/6 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

28. Example 11 Solve cot π/2 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

29. Example 12 Solve tan π/2 without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

30. Your Turn Solve cot 4π without a calculator 4/23/2019 1:22 AM
6.4B - The Unit Circle

31. Example 13
Write the expression without any decimals, 𝐬𝐢𝐧 𝝅 𝟔 𝐜𝐨𝐬 𝝅 𝟐 − 𝐜𝐨𝐬 𝝅 𝟔 𝐬𝐢𝐧 𝝅 𝟐
4/23/2019 1:22 AM
6.4B - The Unit Circle

32. Example 14
Write the expression without any decimals, 𝐬𝐢𝐧 𝟕𝝅 𝟑 𝐜𝐨𝐬 𝟓𝝅 𝟒 + 𝐜𝐨𝐬 𝟕𝝅 𝟑 𝐬𝐢𝐧 𝟓𝝅 𝟒
4/23/2019 1:22 AM
6.4B - The Unit Circle

33. Your Turn
Write the expression without any decimals, 𝐜𝐨𝐬 𝟑𝝅 𝟒 𝐬𝐢𝐧 𝟓𝝅 𝟔 − 𝐬𝐢𝐧 𝟕𝝅 𝟑 𝐜𝐨𝐬 𝟓𝝅 𝟔
4/23/2019 1:22 AM
6.4B - The Unit Circle

34. Assignment Unit Circle Worksheet 1 4/23/2019 1:22 AM
6.4B - The Unit Circle