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Apr 19, 2023Apr 19, 2023
Area of ANY TriangleArea of ANY Triangle
AA
BB
CC
aa
bb
cc
The area of ANY triangle can be found The area of ANY triangle can be found by the following formula.by the following formula.
Another versionAnother version
Another versionAnother version
Key feature Key feature
To find the areaTo find the areayou need to knowing you need to knowing
2 sides and the angle in 2 sides and the angle in between (SAS)between (SAS)
AbcArea sin21
BacArea sin21
CabArea sin21
If you know A, b and cIf you know A, b and cIf you know B, a and cIf you know B, a and cIf you know C, a and bIf you know C, a and b
Area of ANY TriangleArea of ANY Triangle
AA
BB
CC
AA
20cm20cmBB
25cm25cm
CCcc
Example : Find the area of the triangle.Example : Find the area of the triangle.
The version we use isThe version we use is
3030oo
120 25 sin 30
2oArea
210 25 0.5 125Area cm
CabArea sin21
Area of ANY TriangleArea of ANY Triangle
DD
EE
FF
10cm10cm
8cm8cm
Example : Find the area of the triangle.Example : Find the area of the triangle.
sin1
Area= df E2
The version we use isThe version we use is
6060oo
18 10 sin 60
2oArea
240 0.866 34.64Area cm
The side opposite angle The side opposite angle AA is labelled is labelled aa
The Sine RuleThe Sine Rule
Bsinb
Asin
a
AA
BB
CC
aa
bb
cc
The side opposite angle The side opposite angle BB is labelled is labelled bbThe side opposite angle The side opposite angle CC is labelled is labelled cc
Csinc
The RuleThe Rule
Calculating Sides Using The Sine RuleCalculating Sides Using The Sine Rule
Find the length of Find the length of xx in in this triangle.this triangle.
ox
41sin
o34sin
10 Now cross Now cross multiply.multiply.
oox 41sin1034sin
o
ox
34sin
41sin10
mx 74.11559.0
656.010
Example 1Example 1
PP3434oo
4141oo
xx10m10m
RR
Rr
Pp
sinsinsin
1010
sin 34sin 34°°
xx
sin 41sin 41°°
Find the length of x in this triangle.
ox
133sin
o37sin
10
oox 133sin1037sin
o
ox
37sin
133sin10
602.0731.010
x = 12.14m= 12.14m
Example 210m133o
37o
xD
E
F
Ff
Ee
Dd
sinsinsin
1010
sin 37sin 37°°
xx
sin 133sin 133°°
Calculating Angles Using The Sine RuleCalculating Angles Using The Sine Rule
Example 1.
Find the angle Find the angle aaoo
oasin
45o23sin
38
ooa 23sin45sin38
38
23sin45sin
ooa = 0.463
ooa 6.27463.0sin 1
ao
45m
23o
38m
Z
Y
X
Zz
Yy
Xx
sinsinsin
4545
sin sin aaºº
3838
sin 23sin 23ºº
Cross Cross multiplymultiply
Use sinUse sin-1-1
Example 2.
143o
75m
38m
boFind the size of Find the size of the angle bthe angle boo
obsin
38
oob 143sin38sin75
o143sin
75
75
143sin38sin
oob = 0.305
oob 8.17305.0sin 1
Sine Rule (Bearings)Sine Rule (Bearings)
Consider two radar stations Alpha and Beta. Consider two radar stations Alpha and Beta. Alpha is 140 miles west of Beta. The bearing of an aero plane from Alpha is 140 miles west of Beta. The bearing of an aero plane from Alpha is 032° and from Beta it is 316°. Alpha is 032° and from Beta it is 316°. How far is the aeroplane from Beta? How far is the aeroplane from Beta?
NN
BB AA
3232°°
NN
316316°°140 140 milesmiles
PP
5858°° 4646°°
7676°° We are required to find PBWe are required to find PB
76sin140
46sin
PB
46sin14076sin PB
76sin46sin140 PBmilesPB 104
C
B
AApr 19, 2023Apr 19, 2023
Cosine RuleCosine Rule
a
b
c
The Cosine Rule can be used with ANY triangle The Cosine Rule can be used with ANY triangle
as long as we have been as long as we have been givengiven enough information enough information.
Abccba cos2222
Bcaacb cos2222
c2 a2 b2 2abcos C
Given Given angle Aangle AGiven Given angle Bangle BGiven Given angle Cangle C
Using The Cosine RuleUsing The Cosine RuleExample 1 : Find the unknown side in the triangle below:Example 1 : Find the unknown side in the triangle below:
Identify sides Identify sides a, b, ca, b, c and and angle angle AAoo
aa = = xx bb = = 55 cc = =1212 AAo o == 4343ooWrite down the Cosine Rule Write down the Cosine Rule
for for aa
Substitute valuesSubstitute valuesxx22 = = 5522 ++ 121222 - 2 - 2 xx 5 5 xx 12 cos 43 12 cos 43oo
xx22 = = 81.2881.28 Square root to find “Square root to find “xx”.”.
xx = 9.02m = 9.02m
xx5m5m
12m12m
4343oo
AA BB
CC
aa22 = = bb22 + + cc22 - 2 - 2bcbc cos cos AA
pp22 = = qq22 + + rr22 – 2 – 2pqpq cos P cos P
Example 2Example 2 : :
Find the length of side QRFind the length of side QR
Identify the sides and angle.Identify the sides and angle.p = yp = y rr = 12.2 = 12.2 qq = 17.5 = 17.5 PP = 137= 137oo
Write down Cosine RuleWrite down Cosine Rule for for pp
yy22 = 12.2 = 12.222 + 17.5 + 17.522 – 2 – 2 xx 12.2 12.2 xx 17.5 17.5 xx cos 137 cos 137oo
yy22 = 767.227 = 767.227
yy = 27.7m = 27.7m
Using The Cosine RuleUsing The Cosine Rule
137137oo 17.5 m17.5 m12.2 m12.2 m
yy
PP
QQ RR
SubstituteSubstitute
Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule
The Cosine Rule formula can be rearranged to The Cosine Rule formula can be rearranged to allow us to find the size of an angleallow us to find the size of an angle
bcacb
A2
cos222
This formula is cyclic, depending This formula is cyclic, depending on the angle to be foundon the angle to be found
acbca
B2
cos222
ab
cbaC
2cos
222
Label and identify angles Label and identify angles and sidesand sides
D = D = xxoo dd = 11 = 11 ee = 9 = 9 ff = 16 = 16
Substitute values into the formula.Substitute values into the formula.
cos cos xx == 0.750.75Use cosUse cos-1-1 0.75 to find 0.75 to find xx
xx = 41.4 = 41.4o o
Example 1Example 1 : Calculate the : Calculate the
unknown angle, xunknown angle, xoo . .
Finding Angles Using The Cosine RuleFinding Angles Using The Cosine Rule
DD EE
FF
Write the formula for cos DWrite the formula for cos Dcos D =cos D =e e 22 ++ f f 22 -- d d 22
22efef
cos cos xx = =9922 ++ 161622 -- 111122
2 2 x x 9 9 x x 1616
Example 2: Find the unknown angle in the triangle:Example 2: Find the unknown angle in the triangle:
Write down the formula for cos BWrite down the formula for cos B
Label and identify the Label and identify the sides and angle.sides and angle.
B = yB = yoo aa = 13 = 13 bb = 26 = 26 cc = 15 = 15
The negative tells you the angle is obtuse.The negative tells you the angle is obtuse.
yy = 136.3 = 136.3oo
AA
BB
CC
cos B =cos B =a a 22 ++ c c 22 -- b b 22
22acac
Substitute valuesSubstitute values
cos cos yy == - 0.723- 0.723
cos cos yy = =131322 ++ 151522 -- 262622
2 2 x x 13 13 x x 1515
Use cosUse cos-1-1 -0.723 to find -0.723 to find yy
1.1. Do you know the length of ALL the sides? Do you know the length of ALL the sides?
Cosine Rule or Sine RuleCosine Rule or Sine Rule
How to determine which rule to useHow to determine which rule to use
2.2. Do you know 2 sides and the angle in between? Do you know 2 sides and the angle in between?
SASSASOROR
If YES to either of the questions then Cosine RuleIf YES to either of the questions then Cosine Rule
Otherwise use the Sine RuleOtherwise use the Sine Rule
Two questionsTwo questions
The Sine Rule a b cSinA SinB SinC
Application Problems
25o
15 m AD
The angle of elevation of the top of a building
measured from point A is 25o. At point D which is
15m closer to the building, the angle of elevation is
35o Calculate the height of the building.
T
B
Angle TDA =
145o
Angle DTA =
10o
o o
1525 10
TDSin Sin
o15 2536.5
10Sin
TD mSin
35o
36.5
o3536.5TB
Sin
o36.5 25 0. 93TB Sin m
180 – 35 = 145o
180 – 170 = 10o
The Sine Rule a b cSinA SinB SinC
A
The angle of elevation of the top of a column measured from point A, is 20o. The angle of elevation of the top of the statue is 25o. Find the height of the statue when the measurements are taken 50 m from its base
50 m
Angle BCA =
70o
Angle ACT = Angle ATC =
110o
65o
53.21 m
B
T
C
180 – 110 = 70o 180 – 70 = 110o 180 – 115 = 65o
20o25o
5o
oo 65sin
21.53
5sin
TC
o
o
65sin
5sin21.53 TC
=5.1 m=5.1 m
AC50
20cos o
o20cos
50 AC
m21.53
A fishing boat leaves a harbour (H) and travels due East for 40 miles to a marker buoy (B). At B the boat turns left and sails for 24 miles to a lighthouse (L). It then returns to harbour, a distance of 57 miles.
(a) Make a sketch of the journey.
(b) Find the bearing of the lighthouse from the harbour. (nearest degree)
The Cosine Rule
Application Problems
2 2 2
2b c a
CosAbc
H40 miles
24 miles
B
L
57 miles
A
o20.4A
Bearing Bearing = 90 – 20 = = 90 – 20 = 070070°°
2020°°
NN
40572244057
cos222
A
2 2 2
2b c a
CosAbc
The Cosine Rule a2 = b2 + c2 – 2bcCosA
An AWACS aircraft takes off from RAF Waddington (W) on a navigation exercise. It flies 530 miles North to a point (P) as shown, It then turns left and flies to a point (Q), 670 miles away. Finally it flies back to base, a distance of 520 miles.
Find the bearing of Q from point P.
P
670 miles
W
530 miles
Not to Scale
Q
520 miles6705302520670530
cos222
P
PP = 48.7 = 48.7° (49°)° (49°)
Bearing Bearing = 180 + = 180 + 49 = 49 = 229°229°
1. A town B is 20 km due north of town A and a town C is 15 km north-west of A. Calculate the distance between B and C.
2. Two ships leave port together. One sails on a course of 045° at 9 km/h and the other on a course of 090° at 12 km/h.
After 2h 30 min, how far apart will they be?
3. From a point O, the point P is 3 km distant on a bearing of 040° and the point Q is 5 km distant on a bearing of 123°.
What is the distance between P and Q ?
1. BC2 = 152 + 202 21520cos45° = 225 + 400 6000·7071 = 200·7359
BC = 14·17 km
45°
West
15 km
20 km
North
C
A
B
2. QR2 = 302 + 22·52 23022·5cos45°
= 900 + 506·25 954·59 = 451·6558
QR = 21·25 km
NorthQ
P
22·5 km
30 km
45°R
3. PQ2 = 32 + 52 235cos83° = 9 + 25 300·12187
= 30·344PQ = 5·51 km
Cosine Rule
Bearings problems
Cosine Rule- Bearings:-Two Ships.
In each example the distances and bearings of two ships from a port are given. Use the cosine rule to find the
distance between the two ships.
1. Ship1 [ 74km, 053° ] ; Ship2 [ 104km, 112° ]
Port
Ship1
Ship2
North
Abccba cos2222
5353°°
112112°°
5959°°
Bearings Problems
3. A ship sails 80 km on a bearing of 060° from its home port. It then sails 93 km on a bearing of 134°. How far is it now from its home port ?
4. Glasgow airport is 73 km from Edinburgh airport and lies to the west of Edinburgh airport. The bearing of an aeroplane from Glasgow airport is 040° while its bearing from Edinburgh airport is 300°. How far is the aeroplane from each airport ?
