Transcript C2 - Chapter 6 - Radians

C2: Chapter 6 Radians
Dr J Frost ([email protected])
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)
?