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GBK Precalculus Jordan Johnson

GBK Precalculus Jordan Johnson. Today’s agenda. Greetings Intro Review / Submit HW Lesson : Sums & Products w/Similar Periods Classwork Homework Clean-up. On sinusoids of different periods. The text has addressed how to graph the product of sinusoids of “greatly different periods ”…

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GBK Precalculus Jordan Johnson

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  1. GBK PrecalculusJordan Johnson

  2. Today’s agenda • Greetings • Intro • Review / Submit HW • Lesson: Sums & Products w/Similar Periods • Classwork • Homework • Clean-up

  3. On sinusoids of different periods • The text has addressed how to graph the product of sinusoids of “greatly different periods”… • …but not how to deal with nearly-equal periods.

  4. Sums and Products • Write an equation for y as a product of two sinusoids. Confirm with your grapher that the equation is correct. • What are the periods of the two sinusoids?

  5. Sums and Products • On the same screen, plot the graph of: • y = cos 10 + cos 8 • What do you notice?

  6. Sums and Products • What are the periods of the two sinusoids in y = cos 10 + cos 8? • How are the arguments to cosine in this equation related to those of the one you found earlier?

  7. Sums and Products: Problem • Prove algebraically that the two equations are equivalent. • Use the identities you already know.

  8. Sums and Products: Problem • Prove algebraically that the two equations are equivalent. • Notice: • cos θ cos 9θ appears in the angle sum formula for cos(A ± B), where A and B are θand 9θ.

  9. Sums and Products: Problem • Prove algebraically that the two equations are equivalent. • A trick: • cos 10θ + cos 8θ = cos(9θ + θ) + cos(9θ – θ) = cos 9θ cos θ – sin 9θ sin θ + cos 9θ cos θ + sin 9θ sin θ = 2 cos 9θ cos θ

  10. Formulas • We know: • sin(A + B) = sin A cos B + cos A sin B • sin(A – B) = sin A cos B – cos A sin B • By addition/subtraction, we can derive: • sin(A + B) + sin(A – B) = 2 sin A cos B • sin(A + B) – sin(A – B) = 2 cos A sin B

  11. Formulas • Likewise for cosine: • cos(A + B) = cos A cos B – sin A sin B • cos(A – B) = cos A cos B + sin A sin B • By addition/subtraction: • cos(A + B) + cos(A – B) = 2 cos A cos B • cos(A + B) – cos(A – B) = -2 sin A sin B • cos(A – B) – cos(A + B) = 2 sin A sin B

  12. Classwork / HW • From Section 5-4: • Exercises 7, 9, 11. • Bonus: Exercise 12. • Musical Harmony problems.

  13. Clean-up / Reminders • Pick up all trash / items. • Push in chairs (at front and side tables). • See you tomorrow!

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