C 2 Chapter 6 Radians Dr J Frost

Radians So far you’ve used degrees as the unit to measure angles. But outside

Converting between radians and degrees 180, ? 180° = Note that typically with radians,

Be able to convert these without even thinking. . . 45° = /4? 60°

Using your calculator When using sin/cos/tan, you need to make sure your calculator is

Arc length r Radians Degrees From before, we know that 1 radian gives an

Segment Area r r Radians ? This is just a sector with a triangle

Exam Question a) b) c) ? DC = 3 cm. Using cosine rule, BC

Exam Question 6. 3 ? 28. 35? 7. 58 (not ? 7. 59) 5.

Exercise 6 D Q 2 M N The diagram shows a minor sector OMN

Exercises Exercise 6 C – Page 87 Q 1 a, c, e, 2, 6,

Only 1 in 36 candidates (across the country) got this question fully correct. ?

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C 2: Chapter 6 Radians Dr J Frost (jfrost@tiffin. kingston. sch. uk) www. drfrostmaths. com Last modified: 7 th October 2015

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 360 th of a rotation around a full circle. This is a somewhat arbitrary definition! Click to Start Degree Bromanimation r 1 c 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° = ? 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? 60° = /3? ? 270° = 3 /2 ? 120° = 2 /3 30° = /6? ? 135° = 3 /4 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) c) ? DC = 3 cm. Using cosine rule, BC = 7. 09 cm. And from part ? (a), BD = 5. 6 cm. So perimeter is 15. 7 cm. ?

Exam Question 6. 3 ? 28. 35? 7. 58 (not ? 7. 59) 5. 76 ?

Exercise 6 D Q 2 M N The diagram shows a minor sector OMN of a circle centre O and radius r cm. The perimeter of the sector is 100 cm and the area of the sector is A cm 2. a) Show that A = 50 r – r 2. r cm O Using the information provided: ? We need 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 = 25 cm, and thus the area is (50 x 25) – 252 = 625 cm 2

Exercises Exercise 6 C – Page 87 Q 1 a, c, e, 2, 6, 8, 9, 14 Exercise 6 D – Page 89 (97 in new book) 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) ?