2. Using Fundamental Identities
Sections 5.1 and 5.2

3. OBJECTIVE
Recognize and write the fundamental trig identities. Use the fundamental trig identities to evaluate trig functions, simplify trig expressions, and re-write trig expressions. Verify Trig Identities.

4. Trig Identities can be used to Rewrite Trig Expressions.
RELEVANCE
Trig Identities can be used to Rewrite Trig Expressions.

5. Fundamental Trigonometric Identities
Section 5.1

6. Identities – Prior Knowledge
Reciprocal Identities
Quotient Identities
𝐭𝐚𝐧 𝒖= 𝐬𝐢𝐧 𝒖 𝐜𝐨𝐬 𝒖 𝒄𝒐𝒕 𝒖= 𝒄𝒐𝒔 𝒖 𝒔𝒊𝒏 𝒖
𝒔𝒊𝒏 𝒖= 𝟏 𝒄𝒔𝒄 𝒖 𝒄𝒔𝒄 𝒖= 𝟏 𝒔𝒊𝒏 𝒖 𝒄𝒐𝒔 𝒖= 𝟏 𝐬𝐞𝐜 𝒖 𝒔𝒆𝒄 𝒖= 𝟏 𝐜𝐨𝐬 𝒖 𝐭𝐚𝐧 𝒖= 𝟏 𝐜𝐨𝐭 𝒖 𝐜𝐨𝐭 𝒖= 𝟏 𝐭𝐚𝐧 𝒖

7. 𝑮𝒊𝒗𝒆𝒏: 𝐜𝐨𝐭 𝒖=−𝟓 𝒂𝒏𝒅 𝐬𝐢𝐧 𝒖= 𝟐𝟔 𝟐𝟔
𝑭𝒊𝒏𝒅: 𝐜𝐨𝐬 𝒖.

8. 𝑮𝒊𝒗𝒆𝒏: 𝒔𝒆𝒄 𝒖=− 𝟑 𝟐 𝒂𝒏𝒅 𝐬𝐢𝐧 𝒖= 𝟓 𝟑
𝑭𝒊𝒏𝒅: 𝐭𝐚𝐧 𝒖.

9. 𝑮𝒊𝒗𝒆𝒏: 𝐜𝐬𝐜 𝒖=−𝟐 𝒂𝒏𝒅 𝐭𝐚𝐧 𝒖= 𝟑 𝟑
𝑭𝒊𝒏𝒅: 𝑨𝒍𝒍 𝟔 𝑻𝒓𝒊𝒈 𝑹𝒂𝒕𝒊𝒐𝒔.

10. Pythagorean Identities
Pythagorean Identity 1 - Sin and Cos
Pythagorean Identity 2 - Tan and Sec
Pythagorean Identities 3 - Cot and Csc

11. Pythagorean Identities
𝒔𝒊𝒏 𝟐 𝒖+ 𝒄𝒐𝒔 𝟐 𝒖=𝟏
𝟏+ 𝒄𝒐𝒕 𝟐 𝒖= 𝒄𝒔𝒄 𝟐 𝒖
𝒕𝒂𝒏 𝟐 𝒖+𝟏= 𝒔𝒆𝒄 𝟐 𝒖

12. 𝑮𝒊𝒗𝒆𝒏: 𝒔𝒆𝒄 𝒖=− 𝟑 𝟐 𝒂𝒏𝒅 𝒕𝒂𝒏 𝒖>𝟎
𝑭𝒊𝒏𝒅: 𝑨𝒍𝒍 𝟔 𝑻𝒓𝒊𝒈 𝑹𝒂𝒕𝒊𝒐𝒔.

13. 𝑮𝒊𝒗𝒆𝒏: 𝒄𝒔𝒄 𝒖=− 𝟓 𝟑 𝒂𝒏𝒅 𝒄𝒐𝒔 𝒖>𝟎
𝑭𝒊𝒏𝒅: 𝑨𝒍𝒍 𝟔 𝑻𝒓𝒊𝒈 𝑹𝒂𝒕𝒊𝒐𝒔.

14. Cofunction Identities:
Relationship of Cofunctions
What type of relationship do you notice between cofunctions?

15. Cofunction Identities:
𝒔𝒊𝒏 𝝅 𝟐 −𝒖 = 𝐜𝐨𝐬 𝒖
𝒄𝒐𝒔 𝝅 𝟐 −𝒖 = 𝐬𝐢𝐧 𝒖
𝒕𝒂𝒏 𝝅 𝟐 −𝒖 = 𝐜𝐨𝐭 𝒖
𝐜𝐨𝐭 𝒖= 𝝅 𝟐 −𝒖 = 𝐭𝐚𝐧 𝒖
𝒔𝒆𝒄 𝝅 𝟐 −𝒖 = 𝐜𝐬𝐜 𝒖
𝒄𝒔𝒄 𝝅 𝟐 −𝒖 = 𝐬𝐞𝐜 𝒖

16. 𝑮𝒊𝒗𝒆𝒏: 𝐜𝐨𝒔 𝝅 𝟐 −𝒖 = 𝟑 𝟓 𝒂𝒏𝒅 𝒄𝒐𝒔 𝒖= 𝟒 𝟓
𝑭𝒊𝒏𝒅: 𝑨𝒍𝒍 𝟔 𝑻𝒓𝒊𝒈 𝑹𝒂𝒕𝒊𝒐𝒔.

17. Even/Odd Identities Trig Function Check for Odd/Even Odd of Even sin 𝑢
Explore Even and Odd Functions
Trig Function
Check for Odd/Even
Odd of Even
sin 𝑢
𝑠𝑖𝑛 −𝑢 =− sin 𝑢
ODD
csc 𝑢
𝑐𝑠𝑐 −𝑢 =− csc 𝑢
cos 𝑢
𝑐𝑜𝑠 −𝑢 = cos 𝑢
EVEN
sec 𝑢
𝑠𝑒𝑐 −𝑢 = sec 𝑢
tan 𝑢
𝑡𝑎𝑛 −𝑢 =− tan 𝑢
cot 𝑢
𝑐𝑜𝑡 −𝑢 =− cot 𝑢

18. 𝑮𝒊𝒗𝒆𝒏: 𝐜𝐬𝐜 −𝒖 =−𝟓 𝒂𝒏𝒅 𝒄𝒐𝒔 𝒖= 𝟐𝟒 𝟓
𝑭𝒊𝒏𝒅: 𝑨𝒍𝒍 𝟔 𝑻𝒓𝒊𝒈 𝑹𝒂𝒕𝒊𝒐𝒔.

19. Simplifying Trig Expressions
Identities
Rewrite in Sin and Cos
Factor
Rewrite so it is NOT in Fractional Form
(Multiply By Conjugate)
Substitution
Multiply by “1”

20. 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚:
𝒄𝒐𝒔𝜽 𝒕𝒂𝒏𝜽

21. 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚:
𝒔𝒆𝒄𝜽 𝒄𝒔𝒄𝜽

22. 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚:
𝐬𝐢𝐧 𝒙− 𝐬𝐢𝐧 𝒙 𝒄𝒐𝒔 𝟐 𝒙

23. 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚:
𝒔𝒊𝒏𝜽+𝒄𝒐𝒕𝜽𝒄𝒐𝒔𝜽

24. Using the trig Identities:
Using the Unit Circle:

25. 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚:
𝒔𝒆𝒄𝜽 𝒄𝒔𝒄𝜽

26. 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚:
𝒄𝒐𝒔𝜽𝒔𝒆𝒄𝜽− 𝒄𝒐𝒔 𝟐 𝜽

27. 𝐜𝐨𝐭 𝝅 𝟐 −𝒖 𝒄𝒐𝒔 𝒖
𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚:

28. Verify Trig Equations
Section 5.1

29. 𝐬𝐢𝐧 𝒙 𝟏+ 𝐜𝐨𝐬 𝒙 + 𝐜𝐨𝐬 𝒙 𝐬𝐢𝐧 𝒙 = 𝐜𝐬𝐜 𝒙
𝑽𝒆𝒓𝒊𝒇𝒚:

30. 𝑽𝒆𝒓𝒊𝒇𝒚:
𝒔𝒆𝒄𝜽−𝒕𝒂𝒏𝜽 𝒄𝒔𝒄𝜽+𝟏 =𝒄𝒐𝒕𝜽

31. 𝒄𝒐𝒕 𝝅 𝟐 −𝒖 𝐜𝐨𝐬 𝒖= 𝐬𝐢𝐧 𝒖
𝑽𝒆𝒓𝒊𝒇𝒚:

32. 𝑨𝒅𝒅 𝒐𝒓 𝑺𝒖𝒃𝒕𝒓𝒂𝒄𝒕…..𝒕𝒉𝒆𝒏 𝑺𝒊𝒎𝒑𝒍𝒊𝒇𝒚
𝟏 𝒔𝒆𝒄𝜽+𝟏 − 𝟏 𝒔𝒆𝒄𝜽−𝟏

33. 𝑹𝒆𝒘𝒓𝒊𝒕𝒆 𝒔𝒐 𝒕𝒉𝒂𝒕 𝒊𝒕’𝒔 𝒏𝒐𝒕 𝒊𝒏 𝒇𝒓𝒂𝒄𝒕𝒊𝒐𝒏𝒂𝒍 𝒇𝒐𝒓𝒎:
𝟓 𝐭𝐚𝐧 𝒙+ 𝐬𝐞𝐜 𝒙

34. Homework
𝑾𝒐𝒓𝒌𝒔𝒉𝒆𝒆𝒕 𝟏 – 𝟒𝟎 (𝑶𝑫𝑫)