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Calculus Day 1 Recall from last year  one full rotation = 360 0

Calculus Day 1 Recall from last year  one full rotation = 360 0 Which we now also know = 2π radians Because of this little fact  we can obtain a lot of little special angles and their radian measure equivalence 30 0 = π 45 0 = π 60 0 = π

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Calculus Day 1 Recall from last year  one full rotation = 360 0

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  1. Calculus Day 1 Recall from last year  one full rotation = 3600 Which we now also know = 2π radians Because of this little fact  we can obtain a lot of little special angles and their radian measure equivalence 300 = π 450 = π 600 = π 6 4 3 From these basic special angles you should now be able to fill out the rest of the chart: 900 = 1200 = 1350 = 1500 = 1800 = 2100 = 2250 = 2400 = 2700 = 3000 = 3150 = 3300 = Two important conversions factors are: To convert from radian measure to degree measure: Multiply by180 π But if going from degree’s to radian, we multiply by π 180

  2. Angles of rotation in a circle form a sector of a circle. In a sector three concepts come to mind: The radius = r, the measure of the central angle = θ, and the length of the arc  called the arc length = s. • So keeping all these in mind we remember the formula: s = rθ . • Just remember θ must be represented as a radian measure • So give the radian measure θ if: • r = 5 and s = 6 • 2. r = 8 and s = 6 • Now give the radian measure of θ if: • r = 4 and s = 5 • 4. r = 6 and s = 15

  3. Co-terminal angles are any two angles that share the same terminal ray! • List two co-terminal angles one positive and one negative with each of the following; • (Answers should be in the same unit of measure as the angle given!) • 5. 550 6. -750 7. 3π 8. - 2π • 4 3 • When dealing with sectors of a circle, you now must worry about its area. • So K = ½ r s (keep in mind that sometimes you must start with s = r θ) • A sector of acircle has a radius of 6 cm and a central angle 0.5 radians. • Find its arc length and area

  4. 10. A sectorof a circle has area 25 cm2 and a central angle 0.5 radians. Find its radius and arc length. 11. A sector of a circle has perimeter 7 cm and area 3 cm2. Find all possible radii. (Keep in mind  all perimeter means is total distance around. Look at the pictures of sectors on page 264 and decide what are the pieces of a sector you would have to walk to go all the way around. Then come up with a formula for the perimeter.) 12. Do problem #17 from page 266.

  5. Last but not least lets talk about sine and cosine Remember sin = y and cos = x rr Unless of course you can put yourself on the Unit Circle Then sin = y and cos = x 13. If the terminal ray of an angle θ in standard position passes through (- 3, 2), find sin θ and cos θ 14. If θ is a fourth-quadrant angle and sin θ = - 5 , find cos θ. 13 For these last two problems you may want to keep in mind Chief AllSinTanCos

  6. Remember the unit circle and the coordinates we label at each quandrantal angle? • Use those to help you answer: • sin 1800 • 16. cos 1800 • Now do problems #17-20 on page 272.

  7. Every angle of rotation has a ‘reference angle’ To calculate the Reference angle 1st quadrant angle = ‘reference angle 2nd Quadrant angle = 1800 – angle 3rd quadrant angle = 1800 + angle 4th quad angle = 3600 – angle (Keep in mind Chief AllSinTanCos ) Express sin 2150 in terms of a reference angle Express cos 3120 in terms of its reference angle Find the value of each expression to four decimal places: sin 1220 cos 2370 cos 5 sin (-2)

  8. Remember the special triangles and their ratios: • 30 – 60 - 90 45 – 45 – 90 • 1 : √3 : 2 1 : 1 : √2 • Study the list of special angles and the sine and cosine values on page 277 • Now give the exact value of each expression in simplest radical form: • sin 1350 • cos 2400 • 3. cos 5π • 6 • 4. sin (- 5π ) • 3 • Study the graphs of sin θ and cos θ found on page 278 • Do problems #21 and 23 on page 280.

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