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Trigonometric Identities
Section 5.1 Precalculus PreAP/Dual, Revised ©2017 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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Reciprocal & Quotient Identities
2/17/2019 5:46 AM §5.1: Trigonometric Identities
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Pythagorean Identities
Known as: 𝒂 𝟐 + 𝒃 𝟐 = 𝒄 𝟐 “I TASTE SALT” “I CUT CHEESECAKE” 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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Co-Function Identities
2/17/2019 5:46 AM §5.1: Trigonometric Identities
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Pythagorean Identities
2/17/2019 5:46 AM §5.1: Trigonometric Identities
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Pythagorean Identities
2/17/2019 5:46 AM §5.1: Trigonometric Identities
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Pythagorean Identities
2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§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 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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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 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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Reciprocal Identities Theorem
𝐬𝐢𝐧𝜽= 𝐜𝐬𝐜𝜽= 𝐜𝐨𝐬𝜽= 𝐬𝐞𝐜𝜽= 𝐭𝐚𝐧𝜽= 𝐜𝐨𝐭𝜽= 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§5.1: Trigonometric Identities
Quotient Identities 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§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 𝐭𝐚𝐧𝜽 𝐜𝐨𝐬𝜽=𝐬𝐢𝐧𝜽 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§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 𝐬𝐞𝐜𝜽=𝟏 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§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 𝐭𝐚𝐧𝜽=𝐬𝐞𝐜𝜽 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§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 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§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 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§5.1: Trigonometric Identities
Example 4 Use trigonometric identities to prove 𝟎, 𝝅 𝟐 for 𝐜𝐬𝐜 𝜽+𝐜𝐨𝐭 𝜽 𝐜𝐬𝐜 𝜽−𝐜𝐨𝐭 𝜽 =𝟏 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§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 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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§5.1: Trigonometric Identities
Assignment Page all 2/17/2019 5:46 AM §5.1: Trigonometric Identities
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