CalculusDay 1 Recall from last year one full rotation = 360 0 Which we now also know = 2π radians...
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Transcript of CalculusDay 1 Recall from last year one full rotation = 360 0 Which we now also know = 2π radians...
![Page 1: CalculusDay 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.](https://reader036.fdocuments.us/reader036/viewer/2022072006/56649d1b5503460f949f12ac/html5/thumbnails/1.jpg)
Calculus Day 1Recall 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 3From 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 by 180
π But if going from degree’s to radian, we multiply by π 180
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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
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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)
9. A sector of a circle has a radius of 6 cm and a central angle 0.5 radians. Find its arc length and area
![Page 4: CalculusDay 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.](https://reader036.fdocuments.us/reader036/viewer/2022072006/56649d1b5503460f949f12ac/html5/thumbnails/4.jpg)
10. A sector of 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.
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Last but not least lets talk about sine and cosineRemember sin = y and cos = x r r 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. 13For these last two problems you may want to keep in mind Chief AllSinTanCos
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Remember the unit circle and the coordinates we label at each quandrantal angle?
Use those to help you answer:
15.sin 1800
16. cos 1800
Now do problems #17-20 on page 272.
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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)
![Page 8: CalculusDay 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.](https://reader036.fdocuments.us/reader036/viewer/2022072006/56649d1b5503460f949f12ac/html5/thumbnails/8.jpg)
Remember the special triangles and their ratios:
30 – 60 - 90 45 – 45 – 901 : √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:
1. sin 1350
2. 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.