9.1: Identities and Proofs Grade Distribution 2nd 5th A 4 10 B 8 C 2 1 D F 6 5 No Shows 100+ Range 30-97 53-100 Avg 75.14 82.04 11/8/2018 12:30 PM 9.1: Identities and Proofs

Pre-Calculus AB PreAP/Dual, Revised ©2014 9.1: Identities and Proofs Section 9.1 Pre-Calculus AB PreAP/Dual, Revised ©2014 Viet.j.dang@lonestar.edu 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Review How can I prove that 𝟏+𝐬𝐢𝐧𝒙− 𝐬𝐢𝐧 𝟐 𝒙 𝐜𝐨𝐬𝒙 equals 𝐜𝐨𝐬 𝒙+𝐭𝐚𝐧 𝒙? 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Pythagorean Identities: sin2x + cos2x = 1 1 + tan2x = sec2x 1 + cot2x = csc2x 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Quotient Identity 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Reciprocal Identity 11/8/2018 12:30 PM 9.1: Identities and Proofs

Negative Angle Identity 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Pythagorean Identity 1= 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Steps 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 Make sure that A = B Remember, there is MORE THAN ONE WAY to solve an equation 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 1 Verify the identity, sin x cot x = cos x 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 2 Verify the identity, cot x sec x = csc x 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Your Turn Verify the identity, sec y cos y = 1 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 3 Verify the identity, cot2x sin2x + sin2x = 1 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 3 – Another Way Verify the identity, cot2x sin2x + sin2x = 1 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 4 Verify the identity, (sec x – 1) (sec x + 1) = tan2 x 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 5 Verify the identity, –tan2 x = (tan2x + 1) (cos2x – 1) 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Your Turn Verify the identity, tan2x sec2x – tan2x = tan4 x 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 6 Verify the identity, 𝐬𝐢𝐧 𝟐 𝒙 𝐜𝐨𝐬 𝒙 +𝐜𝐨𝐬𝒙=𝐬𝐞𝐜𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 7 Verify the identity, 𝐜𝐬𝐜 𝒙−𝐬𝐢𝐧 𝒙=𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 7 – Another Way Verify the identity, 𝐜𝐬𝐜 𝒙−𝐬𝐢𝐧 𝒙=𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Your Turn Verify the identity, 𝐭𝐚𝐧 𝒙+𝐜𝐨𝐭 𝒙=𝐬𝐞𝐜 𝒙 𝐜𝐬𝐜 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 8 Verify the identity, 𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙 =𝐬𝐢𝐧 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Your Turn Verify the identity, 𝐭𝐚𝐧 𝒙 𝐬𝐞𝐜 𝒙 =𝐬𝐢𝐧 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 9 Verify the identity, 𝐬𝐢𝐧 𝒙 𝐜𝐬𝐜 𝒙−𝐬𝐢𝐧 𝒙 = 𝐜𝐨𝐬 𝟐 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 9 Verify the identity, 𝐜𝐬𝐜 𝒙 𝐬𝐞𝐜 𝒙−𝐭𝐚𝐧 𝒙 𝐬𝐢𝐧 𝒙 =𝐜𝐨𝐭 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Example 9 Verify the identity, 𝐜𝐬𝐜 𝒙 𝐬𝐞𝐜 𝒙−𝐭𝐚𝐧 𝒙 𝐬𝐢𝐧 𝒙 =𝐜𝐨𝐭 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Your Turn Verify the identity, 𝐜𝐬𝐜 𝒙=𝐬𝐢𝐧 𝒙+𝐜𝐨𝐬 𝒙 𝐜𝐨𝐭 𝒙 11/8/2018 12:30 PM 9.1: Identities and Proofs

9.1: Identities and Proofs Assignment Worksheet 11/8/2018 12:30 PM 9.1: Identities and Proofs