Trigonometry Solving Triangles ADJ OPP HYP Two old angels Skipped over heaven Carrying a harp...

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Trigonometry Solving Triangles

Transcript of Trigonometry Solving Triangles ADJ OPP HYP Two old angels Skipped over heaven Carrying a harp...

Page 1: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

Trigonometry

Solving Triangles

Page 2: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

ADJ

OPPHYP

opposite

adjacent

opposite

hypotenuse

adjacent

hypotenusecos

sin

tan Two old angels

Skipped over heaven

Carrying a harp

Solving Triangles

Page 3: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

Trigonometric ratios in surd form

1

1

1

2 30º

45º

45º

60º

Angles are given in radians

π radians = 180ºπ

3= 60º

π

2= 90º

π

4= 45º

π

6= 30º

3

1

3 3

2

3

1tan 60 º =

sin 60 º =

1

2cos 60 º =

tan 30 º =

1

2 sin 30 º =

3

2 cos 30 º =

2

tan 45 º = 1

1

2 sin 45 º =

1

2cos 45 º =

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Page 4: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

c

a b

Cosine Rule

C

A B

a2 = b2 + c2 – 2bccosA

b a

c

b2 = a2 + c2 – 2accosB

c2 = a2 + b2 – 2abcosC

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Page 5: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

By Pythagoras’ Theorem

a2 = (c – x)2 + h2

a2 = c2 – 2cx + x2 + h2

a2 = b2 + c2 – 2cx

Cosine Rulec

a b

a2 = b2 + c2 – 2bccosA

c

b a

x c – x

b2 = x2 + h2

cosx

Ab

cosx b A

A

h

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The Cosine Rule can be used to find a third side of a triangle if you have the other two

sides and the angle between them.

89o13.8

6·2

w

147o

8 11

m

Cosine Rule

6

10

65o

l

Included angle

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Find the unknown side in the triangle below:

l5 m

12 m

43oIdentify sides a,b,c and angle Ao

a = l b = 5 c = 12 A = 43º

Write down the Cosine Rule

Substitute values and find a2

a2 = 52 + 122 – 2 5 12 cos 43o

a2 = 81·28

Take square root of both sides

a = 9·02 m

Examples

a2 = b2 + c2 – 2bccosA

a2 = 25 + 144 – 120(0·731)

Page 8: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

137o17·5 cm

12·2 cm

a2 = 12·22 + 17·52 – ( 2 12·2 17·5 cos 137o )

a2 = 148·84 + 306·25 – ( 427 – 0·731 )

a2 = 455·09 + 312·137

a2 = 767·227 a = 27·7 cm

Find the unknown side in the triangle below:

Examples

a2 = b2 + c2 – 2bccosA

a = ?

b = 12·2

c = 17·5

A = 137º

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20o

x 6

62 = 102 + x2 – (2 10 x cos 20o)

36 = 100 + x2 – 20x( 0·9397)

0 = x2 – 18·79x + 64

Find the two possible values for the unknown side.

Examples

10

a = 6

b = 10

c = x

A = 20ºa2 = b2 + c2 – 2bccosA

2 4

2

b b acx

a

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20o

x 6

0 = x2 – 18·79x + 64

Find the two possible values for the unknown side.

Examples

10

2( 18·79) ( 18·79) 4(1)(64)

2(1)x

18·79 353·2 256

2x

2 4

2

b b acx

a

18·79 9·86

2

14·3 or 4·5x

a = 6

b = 10

c = x

A = 20º

a 1

b –18·79

c 64

Page 11: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

Find the length of the unknown side in the triangles below:

(1)78o

43 cm

31 cmL

(2)

8 m

5·2 m

38o

M

(3) 110o

6·3 cm

8·7 cm

GL = 47.5cm

M = 5·05 m

G = 12.4cm

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c

a b

Sine Rule

C

A Bba

c

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sin sin sin

a b c

A B C

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____________

5

a

cbd

7

50º

82º

(i) Find dc , correct to the nearest cm.

Multiply both sides by sin 50º

dc sin 48º

In the triangle abc, d is a point on [bc].bd = 5 cm, ac = 7cm, bca = 82º and cad = 50º .

48º

=sin 50º

7

= 7·215…= 7 cm

Angle sum of ∆ is 180º

7

= =c

sin C

asin A

bsin B

_____ _____ _____

Sine Rule

__

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(ii) Find ab , correct to the nearest cm.

In the triangle abc, d is a point on [bc].bd = 5 cm, ac = 7cm, bca = 82º and cad = 50º .

5 d

50º

7

+ = 1212

Cosine Rule:

a2 = b2 + c2 – 2bc cosA

ab 2 = 122 + 72 – 2(12)(7)cos82º

= 144 + 49 – 168cos82º

= 169·6189…

= 13·02…

= 13 cm

ab =

169·6189

a

cb

7

82º

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c

a b

Area of triangle

C

A Bba

c

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1Area sin

2ab C

Must be the included angle

Page 16: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

Calculate the area of the triangle shown.

Give your answer correct to one decimal place.

Area of triangle = absin C12

Area = (3)(4) sin 55 12

= 4·9149… 4

cm

3 cm

C must be the included angle= 4·9 cm2

55º

Page 17: Trigonometry Solving Triangles ADJ OPP HYP  Two old angels Skipped over heaven Carrying a harp Solving Triangles.

Find the area of triangle abc, correct to the nearest whole number.

Area of triangle = absin C12

Area = (14)(18·4) sin 70 12

= 121·0324… 18·4

14

c

a b44º

66º

C must be the included angle

|abc | = 180 – 44 – 66 = 70

70º

= 121units2