Inverse Trig Functions

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1 Inverse Trig Functions
Section 5.6 Calculus AP/Dual, Revised ©2018 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Review What is the domain and range of 𝒇 𝒙 =𝐬𝐢𝐧 −𝟏 𝒙 , 𝐜𝐨𝐬 −𝟏 𝒙 , 𝐚𝐧𝐝 𝐭𝐚𝐧 −𝟏 𝒙 ? 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Inverse Sine 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Inverse Cosine 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Inverse Tangent 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Sine Function 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Cosine Function 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Tangent Function 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

9 The Inverse Graph Restrictions
The restricted domain for sine function is from − 𝝅 𝟐 , 𝝅 𝟐 (Quad I and IV) The restricted domain for cosine function is from 𝟎,𝝅 (Quad I and II) The restricted domain for tangent function is from − 𝝅 𝟐 , 𝝅 𝟐 (Quad I and IV excluding − 𝝅 𝟐 & 𝝅 𝟐 ) 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Inverses None of the six basic trig functions have inverse functions. Restrictions of the domain allow them to have inverses, as demonstrated with the horizontal line test 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 1 Without a calculator, solve 𝐚𝐫𝐜𝐜𝐨𝐬 − 𝟑 𝟐 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Your Turn Without a calculator, solve 𝐚𝐫𝐜𝐬𝐢𝐧 − 𝟐 𝟐 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 2 Evaluate 𝐚𝐫𝐜𝐬𝐢𝐧 −𝟎.𝟑𝟗 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 3 Without a calculator, solve 𝐜𝐨𝐬 𝐬𝐢𝐧 −𝟏 − 𝟐 𝟐 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 4 Without a calculator, solve 𝐜𝐨𝐬 𝐬𝐢𝐧 −𝟏 𝟓 𝟏𝟑 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 5 Without a calculator, solve 𝐬𝐞𝐜 𝐭𝐚𝐧 −𝟏 𝟒𝒙 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Your Turn Without a calculator, solve 𝐜𝐨𝐬 𝐜𝐨𝐭 −𝟏 𝒙 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Proof Take the derivative of 𝒚= 𝐬𝐢𝐧 −𝟏 𝒙 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Proof Take the derivative of 𝒚= 𝐬𝐢𝐧 −𝟏 𝒙 . 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

20 Derivative of Inverse Trig Functions
𝒅 𝒅𝒙 𝐬𝐢𝐧 −𝟏 𝒖 = 𝒅𝒖 𝟏− 𝒖 𝟐 𝒅 𝒅𝒙 𝐜𝐨𝐬 −𝟏 𝒖 =− 𝒅𝒖 𝟏− 𝒖 𝟐 𝒅 𝒅𝒙 𝐭𝐚𝐧 −𝟏 𝒖 = 𝒅𝒖 𝟏+ 𝒖 𝟐 𝒅 𝒅𝒙 𝐜𝐨𝐭 −𝟏 𝒖 =− 𝒅𝒖 𝟏+ 𝒖 𝟐 𝒅 𝒅𝒙 𝐬𝐞𝐜 −𝟏 𝒖 = 𝒅𝒖 𝒖 𝒖 𝟐 −𝟏 𝒅 𝒅𝒙 𝐜𝐬𝐜 −𝟏 𝒖 =− 𝒅𝒖 𝒖 𝒖 𝟐 −𝟏 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 6 Solve 𝒅 𝒅𝒕 𝐬𝐢𝐧 −𝟏 𝒕 𝟐 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 7 Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐭𝐚𝐧 𝟑𝒙 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 8 Solve 𝒅 𝒅𝒙 𝟐 𝐬𝐞𝐜 −𝟏 𝒆 𝟐𝒙 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Your Turn Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐜𝐨𝐬 𝒙 𝟐 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 9 An equation for a tangent to the graph of 𝒚=𝐚𝐫𝐜𝐬𝐢𝐧 𝒙 𝟐 at the origin. 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Example 9 An equation for a tangent to the graph of 𝒚=𝐚𝐫𝐜𝐬𝐢𝐧 𝒙 𝟐 at the origin. 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

27 AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐬𝐢𝐧 𝟐𝒙 (A) −𝟏 𝟐 𝟏−𝟒 𝒙 𝟐 (B) −𝟐 𝟒 𝒙 𝟐 −𝟏 (C) 𝟏 𝟐 𝟏−𝟒 𝒙 𝟐 (D) 𝟐 𝟏−𝟒 𝒙 𝟐 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

28 AP Multiple Choice Practice Question 1 (non-calculator)
Solve 𝒅 𝒅𝒙 𝐚𝐫𝐜𝐬𝐢𝐧 𝟐𝒙 Vocabulary Connections and Process Answer and Justifications 4/12/2019 9:11 PM §5.6: Inverse Trig Functions

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Assignment Page 372 3-19 EOO, odd, odd 4/12/2019 9:11 PM §5.6: Inverse Trig Functions