1. Trigonometric Identities
Section 5.1
Precalculus PreAP/Dual, Revised ©2017
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§5.1: Trigonometric Identities

2. Reciprocal & Quotient Identities
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§5.1: Trigonometric Identities

3. Pythagorean Identities
Known as:
𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐
“I TASTE SALT”
“I CUT CHEESECAKE”
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§5.1: Trigonometric Identities

4. Co-Function Identities
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§5.1: Trigonometric Identities

5. Pythagorean Identities
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§5.1: Trigonometric Identities

6. Pythagorean Identities
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§5.1: Trigonometric Identities

7. Pythagorean Identities
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§5.1: Trigonometric Identities

8. §5.1: Trigonometric Identities
Proving Identities
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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§5.1: Trigonometric Identities

9. Steps in Proving Identities
Work at one side at a time
Identify any opportunities to factor an expression, such as adding fractions, squaring a binomial, etc…
Apply whatever sine and cosine functions are applicable
Simplify the equation by using all applicable theorems and identities
CANCEL, CANCEL, CANCEL
At its conclusion, Make sure that 𝑨=𝑨
Remember, there is MORE THAN ONE WAY to solve an equation
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§5.1: Trigonometric Identities

10. Reciprocal Identities Theorem
𝐬𝐢𝐧⁡𝜽= 𝐜𝐬𝐜⁡𝜽=
𝐜𝐨𝐬⁡𝜽= 𝐬𝐞𝐜⁡𝜽=
𝐭𝐚𝐧⁡𝜽= 𝐜𝐨𝐭⁡𝜽=
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§5.1: Trigonometric Identities

11. §5.1: Trigonometric Identities
Quotient Identities
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§5.1: Trigonometric Identities

12. §5.1: Trigonometric Identities
Example 1
Use trigonometric identities to transform the left side of the equation into the right side 𝟎, 𝝅 𝟐 for 𝐭𝐚𝐧⁡𝜽 𝐜𝐨𝐬⁡𝜽=𝐬𝐢𝐧⁡𝜽
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§5.1: Trigonometric Identities

13. §5.1: Trigonometric Identities
Example 2
Use trigonometric identities to transform the left side of the equation into the right side 𝟎, 𝝅 𝟐 for 𝐜𝐨𝐬 𝜽 𝐬𝐞𝐜⁡𝜽=𝟏
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§5.1: Trigonometric Identities

14. §5.1: Trigonometric Identities
Your Turn
Use trigonometric identities to transform the left side of the equation into the right side 𝟎, 𝝅 𝟐 for 𝐜𝐬𝐜 𝜽 𝐭𝐚𝐧⁡𝜽=𝐬𝐞𝐜⁡𝜽
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§5.1: Trigonometric Identities

15. §5.1: Trigonometric Identities
Example 3
Use trigonometric identities to prove 𝟎, 𝝅 𝟐 for 𝐜𝐨𝐭 𝟐 𝜽 ∙ 𝐬𝐢𝐧 𝟐 𝜽+ 𝐬𝐢𝐧 𝟐 𝜽=𝟏
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§5.1: Trigonometric Identities

16. §5.1: Trigonometric Identities
Example 3 (Another Way)
Use trigonometric identities to prove 𝟎, 𝝅 𝟐 for 𝐜𝐨𝐭 𝟐 𝜽 ∙ 𝐬𝐢𝐧 𝟐 𝜽+ 𝐬𝐢𝐧 𝟐 𝜽=𝟏
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§5.1: Trigonometric Identities

17. §5.1: Trigonometric Identities
Example 4
Use trigonometric identities to prove 𝟎, 𝝅 𝟐 for 𝐜𝐬𝐜 𝜽+𝐜𝐨𝐭 𝜽 𝐜𝐬𝐜 𝜽−𝐜𝐨𝐭 𝜽 =𝟏
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§5.1: Trigonometric Identities

18. §5.1: Trigonometric Identities
Your Turn
Use trigonometric identities to prove 𝟎, 𝝅 𝟐 for 𝐜𝐬𝐜 𝜽+𝟏 𝐜𝐬𝐜 𝜽+𝟏 𝐜𝐬𝐜 𝜽−𝟏 = 𝐜𝐨𝐭 𝟐 𝜽
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§5.1: Trigonometric Identities

19. §5.1: Trigonometric Identities
Assignment
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§5.1: Trigonometric Identities