C2: Chapter 6 Radians Dr J Frost (jfrost@tiffin.kingston.sch.uk) Last modified: 7th 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 1° A degree is a 360th of a rotation around a full circle. This is a somewhat arbitrary definition! Click to Start Degree Bromanimation r 1c One radian however is the movement of one radius’ worth around the circumference of the circle. Click to Start Radian Bromanimation Thinking about how many radii around the circumference we can go: 360° = 2 ? rad Converting between radians and degrees 180, ? 180° = 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° = ? 90° = /2? /3 = 60°? 4/15 = 48°? 45° = /4? /6 = 30°? ? 7/8 = 157.5° ? 72° = 1.257 Be able to convert these without even thinking... 45° = /4? 30° = /6? 60° = /3? ? 135° = 3/4 ? 270° = 3/2 ? 120° = 2/3 90° = /2? 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 4 (for radians) Setup Try evaluating this: ? Arc length r Radians Degrees From before, we know that 1 radian gives an arc of 1 radius in length, so... ? ? Sector Area r Radians Degrees ? ? Segment Area r r Radians ? This is just a sector with a triangle cut out. 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) ? Exercise 6D Q2 M N 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. 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 Exercises Exercise 6D – Page 97 Odd questions Only 1 in 36 candidates (across the country) got this question fully correct. (Hint: introduce a variable r and try to form a rightangled triangle) ?