5.2 sum and difference formulas
DESCRIPTION
5.2 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) - PowerPoint PPT PresentationTRANSCRIPT
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°