1. Pre-Calculus PreAP/Dual, Revised ©2014
Basic Trig Identities
Section 6.5
Pre-Calculus PreAP/Dual, Revised ©2014
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6.5 - Basic Trig Identities

2. 6.5 - Basic Trig Identities
Definition
An identity is an equation that is true for all values of the variables for which every term of the equation is defined.
An odd function is a function that is symmetrical at the origin
An even function is a function that is symmetrical through the y-axis
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6.5 - Basic Trig Identities

3. Reciprocal Identities Theorem
sin θ= csc θ=
cos θ= sec θ=
tan θ= cot θ=
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6.5 - Basic Trig Identities

4. 6.5 - Basic Trig Identities
Quotient Identities
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6.5 - Basic Trig Identities

5. 6.5 - Basic Trig Identities
Example 1
Simplify tan t cos t
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6.5 - Basic Trig Identities

6. 6.5 - Basic Trig Identities
Example 2
Simplify cos t sec t
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6.5 - Basic Trig Identities

7. 6.5 - Basic Trig Identities
Your Turn
Simplify csc t tan t
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6.5 - Basic Trig Identities

8. Pythagorean Identities
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6.5 - Basic Trig Identities

9. Pythagorean Identities
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6.5 - Basic Trig Identities

10. Pythagorean Identities
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6.5 - Basic Trig Identities

11. 6.5 - Basic Trig Identities
Example 3
Simplify, tan2t • cos2t + cos2t
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6.5 - Basic Trig Identities

12. 6.5 - Basic Trig Identities
Example 4
Simplify, 𝐬𝐢𝐧 𝜽 𝐜𝐨𝐬 𝜽 + 𝐜𝐨𝐬 𝜽 𝟏+𝐬𝐢𝐧 𝜽
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6.5 - Basic Trig Identities

13. 6.5 - Basic Trig Identities
Example 5
Simplify, 𝐬𝐞𝐜 𝟐 𝒕− 𝐭𝐚𝐧 𝟐 𝒕 𝐭𝐚𝐧 𝟐 𝒕
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6.5 - Basic Trig Identities

14. 6.5 - Basic Trig Identities
Your Turn
Simplify, 𝐭𝐚𝐧 𝒕+ 𝐜𝐨𝐭 𝒕 𝐭𝐚𝐧 𝒕
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6.5 - Basic Trig Identities

15. Negative Reciprocal Identity (AKA) Even-Odd Identity
Reflects across the x-axis
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6.5A - More Trig Identities

16. 6.5A - More Trig Identities
Definition
An odd function is a function that is symmetrical at the origin
sin (–t) = –sin t
tan (–t) = –tan t
An even function is a function that is symmetrical through the y-axis
cos (–t) = cos t
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6.5A - More Trig Identities

17. 6.5A - More Trig Identities
Example 6
Determine whether f(t) = sin t is an odd or even function. Justify answer.
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6.5A - More Trig Identities

18. 6.5A - More Trig Identities
Example 7
Determine whether f(t) = t sin t is an odd or even function. Justify answer.
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6.5A - More Trig Identities

19. 6.5A - More Trig Identities
Your Turn
Determine whether f(t) = t cos t is an odd or even function. Justify answer.
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6.5A - More Trig Identities

20. 6.5A - More Trig Identities
Example 8
Find sin t and tan t when 𝐜𝐨𝐬 𝒕=− 𝟑 𝟏𝟎 within the quadrant in 𝝅<𝒕< 𝟑𝝅 𝟐
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6.5A - More Trig Identities

21. 6.5A - More Trig Identities
Example 9
Find sin t, cos t, and tan t when 𝐬𝐞𝐜 𝒕=𝟐 where 𝐬𝐢𝐧 𝒕<𝟎
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6.5A - More Trig Identities

22. 6.5A - More Trig Identities
Your Turn
Find cos t, and tan t when 𝐬𝐢𝐧 𝒕=− 𝟏 𝟒 where 𝐬𝐞𝐜 𝒕<𝟎
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6.5A - More Trig Identities

23. 6.5 - Basic Trig Identities
Assignment
Page all, 15, 17, 23, odd all
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6.5 - Basic Trig Identities