Applied to

non-right angled triangles

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IntroductionIn Sec 2,you have learnt to apply the

trigonometric ratios to right angled triangles.

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A

hyp

adj

opp

adj

opptanA

hyp

adjcosA

hyp

oppsinA

How can trigonometry be applied to triangles which are not right angled?

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Let’s refer to the triangles below:

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Q

q

p

r

R

P

bA

ac

C

B In ABC The side opposite angle A

is called a. The side opposite angle B

is called b.

In PQR The side opposite angle P

is called p.

And so on

There are two new rules known as the sine rule and cosine rule.

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1. The sine ruleDraw the perpendicular from C to

meet AB at D.

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Using BDC: DC = a sinB.BD

Using ADC: DC = b sinA.

Therefore a sinB = b sinA.

In the same way:

Putting both results together:

The proof needs some changes to deal with obtuse angles.

Example 1Find the length of BC.

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Substitute A = 32o, C = 85o, b = 5.3:

Multiply by sin32o:

A

a

5.3 cm

b

C

B

32o

85o

Example 2Find the size of angle Z.

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X

Y

y

20 cm

17 cm

Z

110o

Substitute: X= 110o, x = 20, z = 17:

Multiply by 17:

leading to

leading to (or , impossible here).

Avoid finding the largest angle as you probably do not know whether you want the acute or the obtuse angle.

The largest angle is opposite the longest side, the smallest angle opposite the shortest side.

When there are two possible angles they add up to 180 degrees since sin x = sin (180-x).

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Points to note when using the sine rule to find an angle:

2. The cosine rule

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A

a

b

c

C

B

Proof of the cosine ruleApplying Pythagoras’ Theorem

to ADC gives: h2 = b2 – x2

Applying Pythagoras’ Theorem to DBC gives:

a2 = h2 + (c – x)2

= h2 + c2 – 2cx + x2. Substituting from equation into equation k gives:

a2 = b2 – x2 + c2 – 2cx + x2

= b2 + c2 – 2cx. Using DAC again: x = bcosA .

Substituting this into gives: a2 = b2 + c2 – 2cb cosA .

i.e.

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cA

ab

C

B

a2 = b2 + c2 – 2bc

cosA

D

h

c – xx

Again the proof needs some changes to deal with obtuse angles.

Press to skip proof

Cosine rule can be used in two

one for finding a side,one for finding an angle.

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The cosine rule for finding a side.

The formula starts and ends with the same letter, one lower case, one capital.

The square of a side is the sum of the squares of the other 2 sides minus twice the product of the 2 known sides and the cosine of the angle between them.

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bA

ac

C

B

The cosine rule for finding an angle.

The cosine of an angle of a triangle is the sum of the squares of the sides forming the angle minus the square of the side opposite the angle all divided by twice the product of first two sides.

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A

a

b

c

C

B

Starting from:

Add 2bc cosA and subtract a2 getting

Divide both sides by 2bc:

P

p

r

q

Q

R

Example 3Find the length of AC.

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Ab

8.4 cm11.2 cm

C

B

79o

Substitute B = 79o, a = 8.4, c = 11.2

into

(Show what you are calculating, but you do not need to show intermediate results.)

Example 4Find the size of angle D.

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R

5 cm

10 cm

7 cm

M

D

Substitute d = 10, r = 5, m = 7

The cosine rule automatically takes care of obtuse angles.

into

getting

leading to

There is no need to show intermediate

results.

How do I know whether to use the sine rule or the cosine rule?To use the sine rule you need to know an angle

and the side opposite it.

You can use it to find a side (opposite a second known angle) or an angle (opposite a second known side).

To use the cosine rule you need to know either two sides and the included angle or all three sides.

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