9.2: Sum and Differences of Trig Functions Homework 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

Sum and Differences Section 9.2 Pre-Calculus AB PreAP/Dual, Revised ©2014 Viet.dang@humble.k12.tx.us 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Question 1: What is Cosine 45? Question 2: What is Cosine 30? Question 3: What is Cosine 15? 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

Addition and Subtraction Identities 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Consider these angles a and b and the distance in the following unit circle is equal. How can one measure distance? 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Consider these two angles a and b and the distance in the following unit circle is equal. What happens when the point is moved towards the x-axis? 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Consider these two angles a and b and the distance in the following two unit circles are equal. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Consider these two angles a and b and the distance in the following two unit circles are equal. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Consider these two angles a and b and the distance in the following two unit circles are equal. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Consider these two angles a and b and the distance in the following two unit circles are equal. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Consider these two angles a and b and the distance in the following two unit circles are equal. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Continue to prove the identity. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Identity Continue to prove the identity. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Steps A. Using the Addition and Subtraction Identities Identify the parts of the given Utilize any parts of the addition/subtraction identities; BREAK IT DOWN Convert into the unit circle, if possible. If not, draw a picture. Simplify B. Hints Sine: SIGNS STAY THE SAME = Sine and functions alternate Cosine: Cosine = CHANGE and functions use the opposite sign and do not alternate functions 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 1 Find the exact value of cos 15 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Earlier… Question 1: What is Cosine 45? Question 2: What is Cosine 30? Question 3: What is Cosine 15? 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Earlier… Find the exact value of cos 15 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 2 Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator. 75° ?? 30° 45° 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 2 Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 2 – Another Way Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator. 120° 75° ?? 45° 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 2 – Another Way Find the exact value of 𝐬𝐢𝐧 𝟓𝝅 𝟏𝟐 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 3 Find the exact value of 𝐜𝐨𝐬 𝟏𝟏𝝅 𝟏𝟐 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Your Turn Find the exact value of 𝐬𝐢𝐧 𝟕𝝅 𝟏𝟐 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions The Conjugate Rule: No RADICAL in the denominator Conjugate is the complex number’s opposite sign When we multiply a compound equation with a radical, we must multiply the OPPOSITE to get rid of the radical to both the NUMERATOR (TOP) and DENOMINATOR (BOTTOM) 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Review Example Rationalize Denominator & Simplify 1 4+ 3 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Review Example Rationalize Denominator & Simplify 1 4+ 3 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Review Example Rationalize Denominator & Simplify 1 4+ 3 We try to get rid of the radicals on the bottom so the bottom will be even With the CONJUGATE 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 4 Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator. 105° 60° 45° 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 4 Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 4 Find the exact value of 𝐭𝐚𝐧 𝟕𝝅 𝟏𝟐 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Your Turn Find the exact value of 𝐭𝐚𝐧 𝟏𝟏𝝅 𝟏𝟐 without a calculator. 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

Steps in Triangles using Sum/Differences Draw the triangle(s) Identify the missing side Determine all three main trig functions Identify and plug into formula 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 5 Given 𝐬𝐢𝐧 𝒙= 𝟒 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅 and 𝐬𝐢𝐧 𝒚=− 𝟓 𝟏𝟑 from 𝝅≤𝒚≤ 𝟑𝝅 𝟐 , identify the value of cos(x + y). 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 5 Given 𝐬𝐢𝐧 𝒙= 𝟒 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅 and 𝐬𝐢𝐧 𝒚=− 𝟓 𝟏𝟑 from 𝝅≤𝒚≤ 𝟑𝝅 𝟐 , identify the value of cos(x + y). 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Example 6 Given 𝐬𝐢𝐧 𝒙= 𝟓 𝟏𝟑 from 𝝅 𝟐 ≤𝒙≤𝝅 and 𝐜𝐨𝐬 𝒚=− 𝟑 𝟓 from 𝝅 𝟐 ≤𝒙≤𝝅, identify the value of tan(x + y). 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Your Turn Given 𝐬𝐢𝐧 𝒙=− 𝟕 𝟐𝟓 from 𝝅≤𝒙≤ 𝟑𝝅 𝟐 and 𝐜𝐨𝐬 𝒚=− 𝟒 𝟓 from𝝅≤𝒙≤ 𝟑𝝅 𝟐 , identify the value of tan(x – y). 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions

9.2: Sum and Differences of Trig Functions Assignment Worksheet 12/26/2018 5:43 PM 9.2: Sum and Differences of Trig Functions