Precalculus PreAP/Dual, Revised ©2017 §5.5: Double Angles Identity Section 5.5 Precalculus PreAP/Dual, Revised ©2017 viet.dang@humbleisd.net 8/2/2019 8:53 AM §5.5: Double Angles Identity

Double Angle Identities (fill in chart) 𝐬𝐢𝐧 𝟐𝜽= 𝟐 𝐬𝐢𝐧𝜽 𝐜𝐨𝐬𝜽 𝐜𝐨𝐬 𝟐𝜽= 𝐜𝐨𝐬 𝟐 𝜽− 𝐬𝐢𝐧 𝟐 𝜽 𝟐 𝐜𝐨𝐬 𝟐 𝜽−𝟏 𝟏−𝟐 𝐬𝐢𝐧 𝟐 𝜽 𝐭𝐚𝐧 𝟐𝜽= 𝟐 𝐭𝐚𝐧𝜽 𝟏− 𝐭𝐚𝐧 𝟐 𝜽 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Proof Proof of 𝐬𝐢𝐧 𝟐𝜽 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Proof Proof of 𝐜𝐨𝐬 𝟐𝜽 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Proof Proof of 𝐜𝐨𝐬 𝟐𝜽 (Second Part) 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Proof Proof of 𝐜𝐨𝐬 𝟐𝜽 (Third Part) 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Proof Proof of 𝐭𝐚𝐧 𝟐𝜽 8/2/2019 8:53 AM §5.5: Double Angles Identity

Steps for Double Angle Identity Simplify the equation Substitute any identities with a double angle identity Factoring 𝑼-Substitution for Trigonometry It is generally assumed if solving for sin, use the sine identity and same goes with cosine identity Make the equation equal to zero and solve 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 1 Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙−𝐬𝐢𝐧𝒙=𝟎 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 1 Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙−𝐬𝐢𝐧𝒙=𝟎 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 2 Use the double angle formula to solve, 𝐬𝐢𝐧 𝟐𝒙+𝐜𝐨𝐬 𝒙=𝟎 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Your Turn Use the double angle formula to solve, 𝟐𝐜𝐨𝐬 𝒙+𝐬𝐢𝐧 𝟐𝒙=𝟎 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 3 Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 3 Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 3 Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 3 Use the double angle formula to solve, 𝒚=𝟒 𝐜𝐨𝐬 𝟐 𝒙−𝟐 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 4 Use the double angle formula to solve, 𝟑 𝐬𝐢𝐧 𝟐 𝒙+𝐜𝐨𝐬 𝟐𝒙−𝟐=𝟎 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Your Turn Use the double angle formula to solve, 𝐜𝐨𝐬 𝟐𝒙+𝐜𝐨𝐬 𝒙=𝟎 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 5 Given 𝐬𝐢𝐧 𝒙= 𝟒 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅 identify the value of 𝐬𝐢𝐧 𝟐𝒙 . 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Example 6 Given 𝐬𝐢𝐧 𝒙=− 𝟕 𝟐𝟓 from 𝝅≤𝒙≤ 𝟑𝝅 𝟐 identify the value of 𝐜𝐨𝐬 𝟐𝒙 . 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Your Turn Given 𝐜𝐨𝐬 𝜽= 𝟓 𝟏𝟑 from 𝟑𝝅 𝟐 ≤𝒙≤𝟐𝝅 identify the value of 𝐭𝐚𝐧 𝟐𝜽 . 8/2/2019 8:53 AM §5.5: Double Angles Identity

§5.5: Double Angles Identity Assignment Worksheet 8/2/2019 8:53 AM §5.5: Double Angles Identity