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5.2 Sum and Difference Formulas

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)

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5.2 Sum and Difference Formulas

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  1. 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?

  2. What’s the point?

  3. 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

  4. Sum and Difference Formulas for Cosines Sum and Difference Formulas for Sines

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

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

  7. 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:

  8. 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.

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

  10. 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 + cosAsinB. sin(A - B) = sin A cosB - cosAsinB. • Sinbad loved to tell people that his and Cosette's signs were the same.

  11. 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) = cosAcosB - sinAsinB. cos(A - B) = cosAcosB + sinAsinB. • 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.

  12. 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

  13. 1. Split 75o into 30o and 45o 2. Use the cosine formula cos(A + B) = cosAcosB - sinAsinB.

  14. cos(A + B) = cosAcosB - sinAsinB. 3. Replace with Trig values cos sin cos sin

  15. cos(A + B) = cosAcosB - sinAsinB.

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

  17. Find the exact value of:

  18. 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.

  19. 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

  20. Verifying Identities

  21. We will work with the left side.

  22. negative What is the formula for cosine?

  23. positive

  24. Sum and Difference Formulas for Tangent

  25. Find tan 105° tan 105° = tan ( 60° + 45°) tan 60° + tan 45° 1 – tan 60° tan 45° =

  26. Find tan 105°

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