Radian Measure in Trigonometry
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KUS objectives
BAT
convert fluently between degrees and radians
BAT
calculate arcs, sectors and segments
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Introduction to Radian measure1
For a circle of radius r, the angle at the centre,
which subtends an arc of length r, is equal to 1 radian
‘If arc AB has length r, then angle AOB is 1 radian
(1
c
or 1 rad)’
As the circumference of the circle is 2
π
r the total
angle is 2
π
radians
Radians are an alternative to degrees. Some
calculations involving circles are easier when Radians
are used, as opposed to degrees.
Converting radians
and degrees
WB 1
Convert the following angles to degrees
a)
b)
c)
Converting degrees
to radians
WB 1
Convert the following angles to radians
d)
e)
f)
Now do skills 249
ARC Length
SECTOR area
The Area of a Sector and Segment
can be worked out using Radians
A
B
O
X
θ
=
=
=
=
=
Multiply by
π
Multiply by r
2
This is the formula’s
usual form
WB 2a arc length
Arc AB of a circle, with centre O
and radius r, subtends an angle of
θ
radians at O. The Perimeter of
sector AOB is 10 cm. Express r in
terms of
θ
.
Length AB = r
θ
WB 3 arc length
Finding the length of an arc is
easier when you use radians
The border of a garden pond consists
of a straight edge AB of length 2.4m,
and a curved part C, as shown in the
diagram below. The curved part is an
arc of a circle, centre O and radius 2m.
Find the length of C.
(We need to work out angle
θ
)
2m
1.2m
(O)
(H)
Inverse sine
Calculator
in Radians
x
Double for
angle AOB
Angle
θ
= 2
π
– 1.287
Angle
θ
= 4.996 rad
O
A
B
WB 4 sector area
In the diagram, the area of the
minor sector AOB is 28.9cm
2
. Given
that angle AOB is 0.8 rad, calculate
the value of r.
Put the
numbers in
½ x 0.8 =
0.4
Divide by
0.4
Square root
WB 5 sector area
A plot of land is in the shape of a
sector of a circle of radius 55m. The
length of fencing that is needed to
enclose the land is 176m. Calculate the
area of the plot of land.
A
B
O
55m
55m
θ
(We need to work out the angle first)
66m
The length of the arc
must be 66m (adds up
to 176 total)
Put the
numbers in
Divide by 55
1.2
c
Put the
numbers in
Now do skills 251
Segment area
Work out the area of a segment using
radians.
O
A
B
r
r
θ
Area of a Segment
Area of Sector AOB – Area of Triangle AOB
Area of Sector AOB
Area of Triangle AOB
Area of the Segment
a = b = r
C =
θ
Factorise
It is not necessary too memorise this formula – use common sense
WB 6 segment area
Calculate the Area of the segment shown in
the diagram below.
Substitute the
numbers in
Work the
parts out
Only round the
final answer
WB 7 segment area
In the diagram AB is the diameter of a
circle of radius r cm, and angle BOC =
θ
radians. Given that the Area of triangle
AOC is 3 times that of the shaded
segment, show that 3
θ
– 4sin
θ
= 0
Area of the shaded segment
Area of triangle AOC
Remember, sin x = sin (180 – x)
a = b = r
Angle =
π
-
θ
AOC = 3 x
shaded segment
Cancel out
1
/
2
r
2
Multiply out the
brackets
Subtract sin
θ
WB 8
Figure 2 shows a plan of a patio. The patio
PQRS
is in the shape of a sector of a circle
with centre
Q
and radius 6 m.
Given that the length of the straight line
PR
is 6
3 m,
(
a
) find the exact size of angle
PQR
in radians.
(
b
) Show that the area of the patio
PQRS
is 12
m
2
.
(
c
) Find the exact area of the triangle
PQR
.
(
d
) Find, in m
2
to 1 decimal place, the area of the
segment
PRS
.
(
e
) Find, in m to 1 decimal place, the perimeter of the
patio
PQRS
.
WB 9
WB 9 exam Q - solution
WB 10
self-assess
One thing learned is –
One thing to improve is –
KUS objectives
BAT
convert fluently between degrees and radians
BAT
calculate arcs, sectors and segments
END
Explore the concept of radian measure, converting between degrees and radians, calculating arc lengths, sectors, and segments, and understanding the importance of radians in trigonometry. Discover formulas and examples illustrating the application of radians in various mathematical calculations.
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Presentation Transcript
Radian Measure
Trigonometry: Radians BAT convert fluently between degrees and radians BAT calculate arcs, sectors and segments KUS objectives Starter: Simplify (without a calculator) sin30 1 + 3 + 3cos60 2sin45 + cos45 1 + 2 tan150 1 + 3 3tan120
Introduction to Radian measure1 For a circle of radius r, the angle at the centre, which subtends an arc of length r, is equal to 1 radian If arc AB has length r, then angle AOB is 1 radian (1cor 1 rad) A r O 1c r r As the circumference of the circle is 2 r the total angle is 2 radians Radians are an alternative to degrees. Some calculations involving circles are easier when Radians are used, as opposed to degrees. B r 2 r 360 180 = 1c 45 135 225 0 30 60 90 120 150 180 210 = = = c c 2 2 c 0? ??
Converting radians and degrees Radians Degrees ??= 180 WB 1 Convert the following angles to degrees ? ? 4? 15 7? 8 a) b) c) 1.8? =7? 8 180 = 1.8 180 =4? 15 180 ? ? ? =1260? 8? =720? 15?= 48 = 103 = 157.5
Converting degrees to radians Degrees Radians ??= 180 WB 1 Convert the following angles to radians d) e) f) 110 350 150 ? ? ? = 110 = 35 = 150 180 180 180 ? =5? 11? 18 =7? 36= 0.611? = 6 ? is often unwritten The superscript Now do skills 249
SECTOR area The Area of a Sector and Segment can be worked out using Radians Area of Sector Total Area Angle at Centre Total Angle = X r A = 2 2 Multiply by O X r X = 2 2 Multiply by r2 2 r B = X 2 1 2r = 2 X This is the formula s usual form
WB 2a arc length Arc AB of a circle, with centre O and radius r, subtends an angle of radians at O. The Perimeter of sector AOB is 10 cm. Express r in terms of . Length AB = r ????????? = ?? + 2? 10 = ?(? + 2) 10 A ? + 2= ? r 10 r r = O ? + 2 r B
WB 3 arc length Calculator in Radians O opp hyp 1.2 2 0.6 Finding the length of an arc is easier when you use radians = sin x (H) x 2m x = sin The border of a garden pond consists of a straight edge AB of length 2.4m, and a curved part C, as shown in the diagram below. The curved part is an arc of a circle, centre O and radius 2m. Find the length of C. C x = sin A B 1.2m (O) Inverse sine = 0.6435 x rad Double for angle AOB = 2 1.287 x rad Angle = 2 1.287 Angle = 4.996 rad = r 2 4.996 9.99 m l l = l = 2m 2m O B A 2.4m (We need to work out angle )
WB 4 sector area In the diagram, the area of the minor sector AOB is 28.9cm2. Given that angle AOB is 0.8 rad, calculate the value of r. 1 2 = r 2 A Put the numbers in 1 2r = 2 28.9 (0.8) x 0.8 = 0.4 = 2 28.9 0.4r A Divide by 0.4 r cm = 2 72.25 r 0.8c O Square root B = 8.5cm r
WB 5 sector area A plot of land is in the shape of a sector of a circle of radius 55m. The length of fencing that is needed to enclose the land is 176m. Calculate the area of the plot of land. The length of the arc must be 66m (adds up to 176 total) = 66 55 = 1.2c = r l Put the numbers in Divide by 55 A 55m 1 2 = r 2 A 66m 1.2c O Put the numbers in 1 55 2 55m = 2 A B (We need to work out the angle first) A= 2 1815 m Now do skills 251
Segment area Work out the area of a segment using radians. Area of a Segment Area of Sector AOB Area of Triangle AOB Area of Sector AOB 1 2 Area of Triangle AOB 1 2 a = b = r C = = r 2 A O = sin A ab C r r 1r sin 2 = 2 A A B Area of the Segment 1r 2 1r 2 1r sin 2 2 2 Factorise 2 ( sin ) It is not necessary too memorise this formula use common sense
WB 6 segment area Calculate the Area of the segment shown in the diagram below. 1r 2 2 ( sin ) Substitute the numbers in 12.5 2 2 sin 3 3 Work the parts out ( ) 0.1811... 3.125 Only round the final answer O 2.5cm 2 0.57 cm 3
WB 7 segment area In the diagram AB is the diameter of a circle of radius r cm, and angle BOC = radians. Given that the Area of triangle AOC is 3 times that of the shaded segment, show that 3 4sin = 0 Area of the shaded segment ( 2r 1 ) 2 sin Area of triangle AOC 1 2ab sin C a = b = r Angle = - 1 2r 2 sin( ) Remember, sin x = sin (180 x) C 1 2r 2 sin AOC = 3 x shaded segment Cancel out 1/2r2 Multiply out the brackets 1 2r 1 2r ( ) = 3 2 2 sin sin A B 0 ( ) sin = 3 sin = sin 3 3sin Subtract sin = 3 4sin 0
WB 8 Figure 2 shows a plan of a patio. The patio PQRS is in the shape of a sector of a circle with centre Q and radius 6 m. Given that the length of the straight line PR is 6 3 m, (a) find the exact size of angle PQR in radians. (b) Show that the area of the patio PQRS is 12 m2. (c) Find the exact area of the triangle PQR. (d) Find, in m2 to 1 decimal place, the area of the segment PRS. (e) Find, in m to 1 decimal place, the perimeter of the patio PQRS.
KUS objectives BAT convert fluently between degrees and radians BAT calculate arcs, sectors and segments self-assess One thing learned is One thing to improve is
