5.2 Sum and Difference FormulasObjective

To develop and use formulas for the trigonometric functions of a sum or difference of two angle measures

Ain’t they a silly bunch?

Yep, they sure are!

What’s the point?What’s the point?

sin(15) = sin(45 - 30)sin(15) = sin(45 - 30)

cos(75) = cos(30 + 45) cos(75) = cos(30 + 45)

Now we just need to know what this Now we just need to know what this means when we use the sum and means when we use the sum and difference formulasdifference formulas

For example…

Sum and Difference Formulas for Cosines

cos cos cos sin sin

cos cos cos sin sin

Sum and Difference Formulas for Sines

5

sin sincos cos

sin sincos cos

Sum and Difference Formulas for Sines

6

cos sincos sin

cos sincos sin

Sum and Difference Formulas for Cosines

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

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

Naturally, when Sinbad was in charge of Naturally, when Sinbad was in charge of their double date, he put himself with their double date, he put himself with Cosette, and he put her sister with his Cosette, and he put her sister with his brother: brother:

sin(A + B) = sin A cosB + cosA sinB. sin(A + B) = sin A cosB + cosA sinB.

sin(A - B) = sin A cosB - cosA sinB. sin(A - B) = sin A cosB - cosA sinB.

Sinbad loved to tell people that his and Sinbad loved to tell people that his and Cosette's signs were the same. Cosette's signs were the same.

However, when Cosette was in charge of the double However, when Cosette was in charge of the double date she placed herself with her sister and put Sinbad date she placed herself with her sister and put Sinbad with his brother. She made sure everyone knew that with his brother. She made sure everyone knew that their signs were their signs were NOTNOT the same: the same:

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

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

Also, notice that Cosette placed herself and her sister Also, notice that Cosette placed herself and her sister BEFOREBEFORE Sinbad and his brother. This detail was Sinbad and his brother. This detail was important to Cosette. She was very snobby, you know. important to Cosette. She was very snobby, you know.

Finding exact values of trig expressions

1. Split the given number into the sum/difference of unit circle values we know

2. Change the problem using the correct formula

3. Simplify by replacing in trig values

1. Split 75o into 30o and 45o

cos 30 45o o

2. Use the cosine formula

0 0cos 30 cos 45 sin 30 sin 45o o

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

3. Replace with Trig values

3 1 2 230 , 45 ,

2 2 2 2o oand

6 2

4 4

6 2

4

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

0 0cos 30 cos 45 sin 30 sin 45o o

sincos cos sin

cos cos105 60 45

cos cos sin sin60 45 60 45

12

22

32

22

24

64

2 64

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:

sin80 cos 20 cos80 sin 20

You will need to know these formulas so let's study them a minute to see the best way to memorize them.

cos cos cos sin sin

cos cos cos sin sin

sin sin cos cos sin

sin sin cos cos sin

cos has same trig functions in first term and in last term, but opposite signs between terms.

opposite

sin has opposite trig functions in each term but same signs between terms.

same

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 values3. Simplify

20

21

Verifying Identities

We will work with the left side.

negative

What is the formula for cosine?

positive

Sum and Difference Formulas for Tangent

tantan tan

tan tan

1

tantan tan

tan tan

1

Find tan 105°

tan 105° = tan ( 60° + 45°)

= tan 60° + tan 45°1 – tan 60° tan 45°

Find tan 105°