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C2: Chapter 6 Radians

C2: Chapter 6 Radians. Dr J Frost (jfrost@tiffin.kingston.sch.uk) . Last modified: 7 th September 2013. Radians. So far you’ve used degrees as the unit to measure angles. But outside geometry, mathematicians pretty much always use radians . r. r. 1 °. 1 c.

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C2: Chapter 6 Radians

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  1. C2: Chapter 6 Radians Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 7thSeptember 2013

  2. Radians So far you’ve used degrees as the unit to measure angles. But outside geometry, mathematicians pretty much always use radians. r r 1° 1c A degree is a 360th of a rotation around a full circle. This is a somewhat arbitrary definition! One radian however is the movement of one radius’ worth around the circumference of the circle. Click to Start Degree Bromanimation Click to Start Radian Bromanimation ? Thinking about how many radii around the circumference we can go: 360° = 2rad

  3. Converting between radians and degrees Note that typically with radians, because it’s considered the ‘preferred choice’ over degrees, we don’t need to write any unit symbol. 180,  ? 180° =   ,  180 ? 180° =  ? 45° = /4 ? 90° = /2 /6 = 30° ? ? /3 = 60° 7/8 = 157.5° ? ? 72° = 1.257 ? 4/15 = 48° ?

  4. Be able to convert these without even thinking... 45° = /4 30° = /6 ? ? 135° = 3/4 60° = /3 ? ? 270° = 3/2 90° = /2 ? ? 120° = 2/3 ?

  5. Using your calculator When using sin/cos/tan, you need to make sure your calculator is in the right mode: degrees or radians. On newer Casio calculators: Shift Setup 4 (for radians) Try evaluating this: ?

  6. Arc length r  Radians Degrees From before, we know that 1 radian gives an arc of 1 radius in length, so... ? ?

  7. Sector Area r  Radians Degrees ? ?

  8. Segment Area This is just a sector with a triangle cut out. r  r Radians ?

  9. Exam Question a) ? b) DC = 3cm. Using cosine rule, BC = 7.09cm. And from part (a), BD = 5.6cm. So perimeter is 15.7cm. ? c) ?

  10. Exercise 6D Q2 The diagram shows a minor sector OMN of a circle centre O and radius r cm. The perimeter of the sector is 100cm and the area of the sector is A cm2. a) Show that A = 50r – r2. M N r cm Using the information provided: ? O We need to get to get rid of  from (2), which we can do by rearranging (1) and substituting it into (2). b) Given that r varies, find the maximum area of the sector OMN. ? So r = 25cm, and thus the area is (50 x 25) – 252 = 625cm2

  11. Exercises Exercise 6D – Page 97 Odd questions

  12. Only 1 in 36 candidates (across the country) got this question fully correct. ? (Hint: introduce a variable r and try to form a right-angled triangle)

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