Inverse Trig Functions Section 4.7 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 5/27/2019 10:40 PM §4.7: Inverse Functions

{( , ), ( , ), ( , )} –3 2 1 6 3 Review Example 1 Determine the inverse of this relation, {(0, –3), (2, 1), and (6, 3)} {( , ), ( , ), ( , )} –3 2 1 6 3 5/27/2019 10:40 PM §4.7: Inverse Functions

Review Example 1 𝒚=𝒙 inverse Mirrored Image 5/27/2019 10:40 PM §4.7: Inverse Functions

To find 𝒇 −𝟏 (𝒙), we have the HORIZONTAL LINE TEST Question If a function is a relation, is an inverse a function as well? inverse To find 𝒇 −𝟏 (𝒙), we have the HORIZONTAL LINE TEST 5/27/2019 10:40 PM §4.7: Inverse Functions

…to Solve the inverse, switch the 𝒙’s and 𝒚’s Review Example 2 Determine the inverse of 𝒚=𝟑𝒙−𝟐 …to Solve the inverse, switch the 𝒙’s and 𝒚’s y = 3x – 2 y = 3 – 2 x x = 3y – 2 x + 2 = 3y 5/27/2019 10:40 PM §4.7: Inverse Functions

Inverse Definitions 𝑺𝒊𝒏 −𝟏 is read as “the inverse sine” Indicates the inverse of the sine function Cosine’s inverse is 𝑪𝒐𝒔 −𝟏 , etc… 𝑺𝒊𝒏 −𝟏 𝒂=𝜽, where 𝐒𝐢𝐧 𝜽=𝒂 There are two ways of writing inverse functions for sine 𝑺𝒊𝒏 −𝟏 𝒂𝒓𝒄𝒔𝒊𝒏 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Sine Graph 5/27/2019 10:40 PM §4.7: Inverse Functions

Sine Function 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Sine 𝒙 𝒚 𝟎 𝝅/𝟐 𝟏 𝝅 𝟑𝝅/𝟐 −𝟏 𝟐𝝅 𝒙 𝒚 𝟎 𝟏 𝝅/𝟐 𝝅 −𝟏 𝟑𝝅/𝟐 𝟐𝝅 5/27/2019 10:40 PM §4.7: Inverse Functions

Cosine Function 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Cosine Graph 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Cosine 𝒙 𝒚 𝟎 𝟏 𝝅/𝟐 𝝅 −𝟏 𝟑𝝅/𝟐 𝟐𝝅 𝒙 𝒚 𝟏 𝟎 𝝅/𝟐 −𝟏 𝝅 𝟑𝝅/𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Tangent Graph 5/27/2019 10:40 PM §4.7: Inverse Functions

Tangent Function 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Tangent 𝒙 𝒚 𝟎 𝑼𝒏𝒅 𝝅/𝟐 −𝟏 𝝅 𝟑𝝅/𝟐 𝟏 𝟐𝝅 𝒙 𝒚 𝑼𝒏𝒅 𝟎 −𝟏 𝝅/𝟐 𝝅 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse Graph Restrictions The restricted domain for sine function is from − 𝝅 𝟐 , 𝝅 𝟐 The restricted domain for cosine function is from 𝟎,𝝅 The restricted domain for tangent function is from − 𝝅 𝟐 , 𝝅 𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Review Find the exact value of 𝐜𝐨𝐬 𝟑𝝅 𝟒 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 1 Find the exact value of 𝐂𝐨𝐬 −𝟏 − 𝟐 𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 2 Find the exact value of 𝐚𝐫𝐜𝐬𝐢𝐧 𝟏 𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Your Turn Find the exact value of 𝐂𝐨𝐬 −𝟏 𝟐 𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 3 Find the exact value of 𝐓𝐚𝐧 −𝟏 𝟑 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 4 Find the exact value of 𝐚𝐫𝐜𝐬𝐢𝐧 − 𝟑 𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Your Turn Find the exact value of 𝐓𝐚𝐧 −𝟏 − 𝟑 𝟑 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Cosecant Graph 5/27/2019 10:40 PM §4.7: Inverse Functions

Cosecant Function 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Secant Graph 5/27/2019 10:40 PM §4.7: Inverse Functions

Secant Function 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse of Cotangent Graph 5/27/2019 10:40 PM §4.7: Inverse Functions

Cotangent Function 5/27/2019 10:40 PM §4.7: Inverse Functions

The Inverse Reciprocal Graphs The restricted domain for cosecant function is from − 𝝅 𝟐 , 𝝅 𝟐 The restricted domain for secant function is from 𝟎,𝝅 The restricted domain for cotangent function is from 𝟎,𝝅 5/27/2019 10:40 PM §4.7: Inverse Functions

Steps Take the reciprocal of the trig function Take the reciprocal of the answer Switch and solve 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 5 Find the exact value of 𝑺𝒆𝒄 −𝟏 − 𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 6 Find the exact value of 𝐚𝐫𝐜𝐬𝐞𝐜 − 𝟐 𝟑 𝟑 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 7 Find the exact value of 𝑪𝒐𝒕 −𝟏 − 𝟑 5/27/2019 10:40 PM §4.7: Inverse Functions

Your Turn Find the exact value of 𝑪𝒔𝒄 −𝟏 − 𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Inverses in the Calculator Make sure that it is in the appropriate mode It is the 𝑻𝒓𝒊𝒈 −𝟏 feature in the graphing calculator For reciprocal functions, type in the 𝑻𝒓𝒊𝒈 −𝟏 𝟏 𝒂𝒏𝒔 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 8 Use the calculator to approximate 𝐚𝐫𝐜𝐭𝐚𝐧(−𝟖.𝟒𝟓) 5/27/2019 10:40 PM §4.7: Inverse Functions

Example 9 Use the calculator to approximate 𝑪𝒐𝒕 −𝟏 𝟑 5/27/2019 10:40 PM §4.7: Inverse Functions

Your Turn Use the calculator to approximate 𝒂𝒓𝒄𝒄𝒔𝒄 −𝟏𝟐 5/27/2019 10:40 PM §4.7: Inverse Functions

Assignment Page 326 5-17 odd (without a calculator), 21-37 EOO (with a calculator), 119-125 all (without), 126-134 all (Round for 4 decimal places) 5/27/2019 10:40 PM §4.7: Inverse Functions