GBK Precalculus Jordan Johnson

1. GBK PrecalculusJordan Johnson

2. Today’s agenda • Greetings • Lesson: Composite Argument Properties • NaQ? • Classwork / Homework • Note: quiz Monday. • Clean-up

3. Reminders • Sine and tangent are odd, cosine is even. • sin(–x) = –sin(x) • cos(–x) = cos(x) • tan(–x) = –tan(x) • Oddness/evenness is called parity. • Complementary argument properties: • sin(θ) = cos(90° –θ), and cos(θ) = sin(90° – θ) • Similar equations relate tan to cot, sec to csc. • True for radian measure too: sin x = cos(/2 – x)

4. Composite Argument Properties • We have proven this formula: • We also derived this formula using matrices: • Now we’re in a position to prove it without them:

5. Composite Arguments to sin • sine of a sum: • sine of a difference:

6. Questions • What is parity? • Why is it relevant here? • What are the complementary argument properties? • Why are they relevant here?

7. Composite Arguments to tan • There’s a formula for tangent of a sum, too:

8. Composite Arguments to tan • Tangent of a difference:

9. Example • Prove: • cos(x – /2) = sin x = cos x cos(/2) + sin x sin(/2) = 0 cos x + 1 sin x = sin x,therefore cos(x – /2) = sin x, q.e.d. • sin( + 30°) + cos( + 60°) = cos 

10. Summary • Write down the composite argument formulas; mark p. 210 in the text, which summarizes them. • Be able to use them • to eliminate sums and differences from arguments • to solve equations • to simplify trig expressions • to prove trig identities

11. Homework • From Section 5-3 (pp. 211-213): • Reading Analysis questions • Q1-Q10 • Problems 1-9 odd, 10, 11-15 odd.(Note: In #10 they call it “proof,” though “graphical evidence” is a more appropriate term.) • Due Monday, 2/7. • Note there’s a quiz Monday, too.

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