6.2 Sum and Difference Formulas Objective To develop and use formulas for the trigonometric functions of a sum or difference of two angle measures Yep, they sure are! Ain’t they a silly bunch?

What’s the point?  

For example… sin(15) = sin(45 - 30) cos(75) = cos(30 + 45) Now we just need to know what this means when we use the sum and difference formulas

Sum and Difference Formulas for Cosines Sum and Difference Formulas for Sines

Sum and Difference Formulas for Sines   sin   cos     cos   sin     sin   cos     cos   sin  

Sum and Difference Formulas for Cosines   cos   cos     sin   sin     cos   cos     sin   sin  

Can you memorize these formulas Can you memorize these formulas? You will have to if you take college trigonometry. Here is a love story to help introduce the trigonometry sum and difference formulas in an interesting way:

As we all know, some of the people to whom we are attracted are not attracted to us. And it is not unusual for a person who has shown interest in us to later lose interest in us. Maybe that is a good thing, because it forces us to date a lot of people and to become more experienced in maintaining relationships.

Anyway, this is the story of Sinbad and Cosette Anyway, this is the story of Sinbad and Cosette. Sinbad loved Cosette, but Cosette did not feel the same way about Sinbad.

Naturally, when Sinbad was in charge of their double date, he put himself with Cosette, and he put her sister with his brother: sin(A + B) = sin A cosB + cosA sinB. sin(A - B) = sin A cosB - cosA sinB. Sinbad loved to tell people that his and Cosette's signs were the same.

cos(A + B) = cosA cosB - sinA sinB. However, when Cosette was in charge of the double date she placed herself with her sister and put Sinbad with his brother. She made sure everyone knew that their signs were NOT the same: cos(A + B) = cosA cosB - sinA sinB. cos(A - B) = cosA cosB + sinA sinB. Also, notice that Cosette placed herself and her sister BEFORE Sinbad and his brother. This detail was important to Cosette. She was very snobby, you know.

Finding exact values of trig expressions Split the given number into the sum/difference of unit circle values we know Change the problem using the correct formula Simplify by replacing in trig values

cos(A + B) = cosA cosB - sinA sinB. 1. Split 75o into 30o and 45o     1. Split 75o into 30o and 45o     2. Use the cosine formula cos(A + B) = cosA cosB - sinA sinB.

You should memorize this You should memorize this. This is a great reference because you can figure out the trig functions of all these angles quickly.    

cos(A + B) = cosA cosB - sinA sinB. 3. Replace with Trig values cos   cos(A + B) = cosA cosB - sinA sinB. 3. Replace with Trig values cos sin cos sin  

cos(A + B) = cosA cosB - sinA sinB.   cos(A + B) = cosA cosB - sinA sinB.

             

  Look at the formulas.  Which one does it match?          

Find the exact value of:    

You will need to know these formulas so let's study them a minute to see the best way to memorize them. opposite cos has same trig functions in first term and in last term, but opposite signs between terms. same sin has opposite trig functions in each term but same signs between terms.

Verifying Identities These three steps are key in verifying identities that require the sum and difference formulas: 1. Write in expanded form 2. Substitute known values 3. Simplify

Verifying Identities      

  We will work with the left side.      

       

Sum and Difference Formulas for Tangent

Find tan 105° tan 105° = tan ( 60° + 45°) tan 60° + tan 45° =    

Find tan 105°