5-may-15 exact values angles greater than 90 o trigonometry useful notation & area of a triangle...

18 Apr 202318 Apr 2023

Exact Values

Angles greater than 90o

TrigonometryTrigonometry

Useful Notation & Area of a triangle

Using Area of Triangle Formula

Cosine Rule Problems

Sine Rule Problems

Mixed Problems

18 Apr 202318 Apr 2023

Starter QuestionsStarter Questions

o

21. Factorise x - 36

2. A car depreciates at 20% each year.

How much is it worth af ter 4 years if it cost

£ 15 000 initially.

3. What sin30 as a f raction.

18 Apr 202318 Apr 2023

Exact ValuesExact Values

Learning IntentionLearning Intention Success CriteriaSuccess Criteria

1.1. Recognise basic triangles Recognise basic triangles and exact values for sin, and exact values for sin, cos and tan 30cos and tan 30oo, 45, 45oo, 60, 60oo . .

1. To build on basic trigonometry values.

2.2. Calculate exact values for Calculate exact values for problems.problems.

22

2

60º

60º

60º 1

60º

230º3

This triangle will provide exact values for

sin, cos and tan 30º and 60º


Some special values of Sin, Cos and Tan are useful left as fractions, We call these exact values

x 0º 30º 45º 60º 90º

Sin xº

Cos xº

Tan xº

½

½

3

3

2

3

20

1

0

1

0


1

3


1 145º

45º

2

Consider the square with sides 1 unit

11

We are now in a position to calculate exact values for sin, cos and tan of 45o

x 0º 30º 45º 60º 90º

Sin xº

Cos xº

Tan xº

½

½

3

3

2

3

20

1

0

1

0


1

3

1 2

1 2

1

18 Apr 202318 Apr 2023

Now try Exercise 1Ch8 (page 94)


18 Apr 202318 Apr 2023


1. Write down the Compound I nterest Formula

and identif y each term.

2. A house increases by 3% each year.

How much is it worth in 5 years if it cost

£ 40 000 initially.

3. What is the .oexact value of sin 45

18 Apr 202318 Apr 2023


Angles Greater than Angles Greater than 9090oo

1. Introduce definition of sine, cosine and tangent over 360o using triangles with the unity circle.

1.1. Find values of sine, cosine Find values of sine, cosine and tangent over the range and tangent over the range 00o o to 360to 360oo..

2.2. Recognise the symmetry Recognise the symmetry and equal values for sine, and equal values for sine, cosine and tangent.cosine and tangent.

Apr 18, 2023Apr 18, 2023 1111

xy

rAngles Greater than Angles Greater than

9090oo

We will now use a new definition to cater for ALL angles.

O x-axis

r

y-axis

y

xAo

New Definitions

siny

Ar

P(x,y)

cosx

Ar

tany

Ax

18 Apr 202318 Apr 2023

TrigonometryTrigonometryAngles over 900

(1.2, 1.6)

53o

The radius line is 2cm.The point (1.2, 1.6).

Find sin cos and tan forthe angle.

1.6sin 53 0.8

2o

1.2cos53 0.6

2o

1.6tan 53 1.33

1.2o

Check answer with

calculator

Example 1

18 Apr 202318 Apr 2023


(-1.8, 0.8)

127o

The radius line is 2cm.The point (-1.8, 0.8).

Find sin cos and tan forthe angle.

0.8sin127 0.4

2o

1.8cos127 0.9

2o

0.8tan 53 0.44

1.8o

Check answer with

calculator

Example 1

1) Sin 135o

2) Cos 150o

3) Tan 135o

4) Sin 225o

5) Cos 270o

What Goes In The Box ?What Goes In The Box ?

Write down the equivalent values of the following in term of the first quadrant (between 0o and 90o):

sin 45o 1) Sin 300o

2) Cos 360o

3) Tan 330o

4) Sin 380o

5) Cos 460o

-cos 45o

-tan 45o

-sin 45o

-cos 90o

- sin 60o

cos 0o

- tan 30o

sin 20o

- cos 80o

18 Apr 202318 Apr 2023



18 Apr 202318 Apr 2023


Extension for unit 2 Trigonometry

GSM Software

All +ve

Sin +ve

Tan +ve Cos +ve

180o - xo

180o + xo 360o - xo

Angles Greater than Angles Greater than 9090oo

(0,1)

(-1,0)

(0,-1)

(1,0)0A

Two diagrams display same data in a different format

18 Apr 202318 Apr 2023


2

o

1. Find the area of the triangle.

2. Factorise x - 4x +3

3. Find the exact value of cos 120 .

8cm

3cm

18 Apr 202318 Apr 2023


Area of a Area of a TriangleTriangle

1. To show the standard way of labelling a triangle.

2. Find the area of a triangle using basic trigonometry knowledge.

1.1. Be able to label a triangle Be able to label a triangle properly. properly.

2.2. Find the area of a triangle Find the area of a triangle using basic trigonometry using basic trigonometry knowledge.knowledge.

18 Apr 202318 Apr 2023

Labelling TrianglesLabelling Triangles

A

B

C

A

aB

b

Cc

Small letters a, b, c refer to distancesCapital letters A, B, C refer to angles

In Mathematics we have a convention for labelling triangles.

F

E

D

F

E

D18 Apr 202318 Apr 2023

Labelling TrianglesLabelling Triangles

d

e

f

Have a go at labelling the following triangle.

18 Apr 202318 Apr 2023

Area of a TriangleArea of a Triangle

A

B

12cm C

10cm

Example 1 : Find the area of the triangle ABC.

50o

(i) Draw in a line from B to AC

(ii) Calculate height BD

D

o BDSin50 =

10oBD = 10 Sin50 = 7.66

2

1

2

0.5 12 7.66 46

Area base height

cm

(iii) Area

7.66cm

18 Apr 202318 Apr 2023

Area of a TriangleArea of a Triangle

Q

P

20cm R

12cm

Example 2 : Find the area of the triangle PQR.

40o

(i) Draw in a line from P to QR

(ii) Calculate height PS

S

o PSSin40 =

10oPS = 12 Sin40 = 7.71

2

1

2

0.5 20 7.71 77.1

Area base height

cm

(iii) Area

7.71cm

18 Apr 202318 Apr 2023


Constructing Pie Constructing Pie ChartsCharts

18 Apr 202318 Apr 2023


2( 3) (4 )1. Multiply out and simplif y

2. Find the volume of a cylinder 15cm in height

and 10cm in diameter.

3. Write down the two values f or sin

that give a value of 0.5

x x

18 Apr 202318 Apr 2023


1.1. Know the formula for the Know the formula for the area of any triangle.area of any triangle.

1. To explain how to use the Area formula for ANY triangle.

Area of ANY TriangleArea of ANY Triangle

2.2. Use formula to find area of Use formula to find area of any triangle given two any triangle given two length and angle in length and angle in between.between.

General Formula forGeneral Formula forArea of ANY TriangleArea of ANY Triangle

Consider the triangle below:

Ao Bo

Co

ab

c

h

Area = ½ x base x height 1

2A c h

What does the sine of Ao equal

sin o hA

b

Change the subject to h. h = b

sinAoSubstitute into the area formula

1sin

2oA c b A

1sin

2oA bc A

18 Apr 202318 Apr 2023


A

B

C

A

aB

b

Cc

The area of ANY triangle can be found by the following formula.

sin1

Area= ab C2

sin1

Area= ac B2

sin1

Area= bc A2

Another version

Another version

Key feature

To find the areayou need to

knowing 2 sides and the

angle in between (SAS)

18 Apr 202318 Apr 2023


A

B

C

A

20cmB

25cm

Cc

Example : Find the area of the triangle.

sinC1

Area= ab2

The version we use is

30o

120 25 sin 30

2oArea

210 25 0.5 125Area cm

18 Apr 202318 Apr 2023


D

E

F

10cm

8cm

Example : Find the area of the triangle.

sin1

Area= df E2

The version we use is

60o

18 10 sin 60

2oArea

240 0.866 34.64Area cm

What Goes In The Box What Goes In The Box ??

Calculate the areas of the triangles below:

(1)

23o

15cm

12.6cm

(2)

71o

5.7m

6.2m

A =36.9cm2

A =16.7m2

Key feature

Remember (SAS)

18 Apr 202318 Apr 2023



18 Apr 202318 Apr 2023


2

1. Multiply out the brackets and simplif y

5(y- 5) - 7(5- y)

2. Find the gradient and the y - intercept

3 f or the line with equation y = 5x -

4

3. Factorise x -100

18 Apr 202318 Apr 2023


1.1. Know how to use the sine Know how to use the sine rule to solve REAL LIFE rule to solve REAL LIFE problems involving problems involving lengths.lengths.

1. To show how to use the sine rule to solve REAL LIFE problems involving finding the length of a side of a triangle .

Sine RuleSine Rule

C

B

A18 Apr 202318 Apr 2023

Sine RuleSine Rule

a

b

c

The Sine Rule can be used with ANY triangle as long as we have been given enough information.

Works for any Triangle

a b c= =

SinA SinB SinC

Deriving the rule

B

C

A

b

c

a

Consider a general triangle ABC.

The Sine Rule

Draw CP perpendicular to BA

P

CPSinB CP aSinB

a

CP

also SinA CP bSinAb

aSinB bSinA

aSinBb

SinA

a bSinA SinB

This can be extended to

a b cSinA SinB SinC

or equivalentlySinA SinB SinCa b c

Calculating Sides Calculating Sides Using The Sine RuleUsing The Sine Rule

10m

34o

41o

a

Match up corresponding sides and angles:

sin 41oa

10

sin 34oNow cross multiply.

sin 34 10sin 41o oa Solve for a.

10sin 41

sin 34

o

oa 10 0.656

11.740.559

a m

Example 1 : Find the length of a in this triangle.

A

B

C

Calculating Sides Calculating Sides Using The Sine Using The Sine

RuleRule

10m133o

37o

d

sin133od

10

sin 37o

sin 37 10sin133o od

10sin133

sin 37

o

od

10 0.731

0.602d

=

12.14m


Now cross multiply.

Solve for d.

Example 2 : Find the length of d in this triangle.

C

D

E

What goes in the Box What goes in the Box ??

Find the unknown side in each of the triangles below:

(1) 12cm

72o

32o

a

(2)

93o

b47o

16mm

a = 6.7cm b =

21.8mm18 Apr 202318 Apr 2023

18 Apr 202318 Apr 2023

Now try Ex 6&7 Ch8 (page 103)

Sine RuleSine Rule

18 Apr 202318 Apr 2023


1. Factorise 9x - 36


3 1 f or the line with equation y = - x +

4 5

3. Write down the two values of cos

1 that give you a value of

2

18 Apr 202318 Apr 2023


1.1. Know how to use the sine Know how to use the sine rule to solve problems rule to solve problems involving angles.involving angles.

1. To show how to use the sine rule to solve problems involving finding an angle of a triangle .

Sine RuleSine Rule

Calculating Angles Calculating Angles

Using The Sine Using The Sine RuleRule

Example 1 :

Find the angle Ao

Ao

45m

23o

38m


45

sin oA 38

sin 23oNow cross multiply:

38sin 45sin 23o oA Solve for sin Ao

45sin 23sin

38

ooA = 0.463 Use sin-1 0.463 to find Ao

1sin 0.463 27.6o oA

Calculating Angles Calculating Angles

Using The Sine Using The Sine RuleRule

143o

75m

38m

Bo

38

sin oB

75sin 38sin143o oB

75

sin143o

38sin143sin

75

ooB = 0.305

1sin 0.305 17.8o oB

Example 2 :

Find the angle Bo


Now cross multiply:

Solve for sin Bo

Use sin-1 0.305 to find Bo

What Goes In The Box What Goes In The Box ??

Calculate the unknown angle in the following:

(1)

14.5m

8.9m

Ao

100o (2)

14.7cm

Bo

14o

12.9cm

Ao = 37.2o

Bo = 16o

18 Apr 202318 Apr 2023

Now try Ex 8 & 9Ch8 (page 106)

Sine RuleSine Rule

18 Apr 202318 Apr 2023


2

1. Find the gradient of the line that passes

through the points ( 1,1) and (9,9).


f or the line with equation y = 1 - x

3. Factorise x - 64

18 Apr 202318 Apr 2023


1.1. Know when to use the Know when to use the cosine rule to solve cosine rule to solve problems.problems.

1. To show when to use the cosine rule to solve problems involving finding the length of a side of a triangle .

Cosine RuleCosine Rule

2. 2. Solve problems that Solve problems that involve finding the length involve finding the length of a side.of a side.

C

B

A18 Apr 202318 Apr 2023


a

b

c

The Cosine Rule can be used with ANY triangle as long as we have been given enough information.


cos2 2 2a =b +c - 2bc A

Deriving the rule

A

B

C

a

b

c

Consider a general triangle ABC. We require a in terms of b, c and A.

Draw BP perpendicular to AC

b

Px b - x

BP2 = a2 – (b – x)2

Also: BP2 = c2 – x2

a2 – (b – x)2 = c2 – x2

a2 – (b2 – 2bx + x2) = c2 – x2

a2 – b2 + 2bx – x2 = c2 – x2

a2 = b2 + c2 – 2bx*

a2 = b2 + c2 – 2bcCosA*Since Cos A = x/c x = cCosA

When A = 90o, CosA = 0 and reduces to a2 = b2 + c2

1

When A > 90o, CosA is positive, a2 > b2 + c2 2

When A < 90o, CosA is negative, a2 > b2 + c2 3

The Cosine Rule

The Cosine Rule generalises Pythagoras’ Theorem and takes care of the 3 possible cases for Angle A.

a2 > b2 + c2

a2 < b2 + c2

a2 = b2 + c2

A

A

A

1

2

3

Pythagoras + a bitPythagoras - a bit

Pythagoras

a2 = b2 + c2 – 2bcCosA

Applying the same method as earlier to the other sides produce similar formulae

for b and c. namely:b2 = a2 + c2 – 2acCosB

c2 = a2 + b2 – 2abCosC

A

B

C

a

b

c

The Cosine Rule

The Cosine rule can be used to find:

1. An unknown side when two sides of the triangle and the included angle are given.

2. An unknown angle when 3 sides are given.

Finding an unknown side.

18 Apr 202318 Apr 2023


How to determine when to use the Cosine Rule.


1. Do you know ALL the lengths.

2. Do you know 2 sides and the angle in between.

SASOR

If YES to any of the questions then Cosine Rule

Otherwise use the Sine Rule

Two questions

Using The Cosine Using The Cosine RuleRule

Example 1 : Find the unknown side in the triangle below: L5m

12m

43o

Identify sides a,b,c and angle Ao

a =

L b =

5 c =

12 Ao = 43o

Write down the Cosine Rule.

Substitute values to find a2.a2 =

52 + 122 - 2 x 5 x 12 cos 43o

a2 =

25 + 144

- (120 x

0.731 )

a2 =

81.28 Square root to find “a”.

a = L = 9.02m


Example 2 :

Find the length of side M.

137o

17.5 m

12.2 m

M

Identify the sides and angle.

a = M

b = 12.2 C = 17.5 Ao = 137o

Write down Cosine Rule

a2 = 12.22 + 17.52 – ( 2 x 12.2 x 17.5 x cos 137o )

a2 = 148.84 + 306.25 – ( 427 x – 0.731 )Notice the two negative

signs.a2 = 455.09 + 312.137

a2 = 767.227

a = M = 27.7m

Using The Cosine Using The Cosine RuleRuleWorks for any Triangle

What Goes In The What Goes In The Box ?Box ?

Find the length of the unknown side in the triangles:

(1)78o

43cm

31cmL

(2)

8m

5.2m

38o

M

L = 47.5cm

M = 5.05m

18 Apr 202318 Apr 2023

Now try Ex 11.1Ch11 (page 142)


18 Apr 202318 Apr 2023


2

1. I f lines have the same gradient

What is special about them.

2. Factorise x +4x -12

3. Find the missing angles.

54o

18 Apr 202318 Apr 2023


1.1. Know when to use the Know when to use the cosine rule to solve cosine rule to solve REAL LIFE problems.problems.

1. To show when to use the cosine rule to solve REAL LIFE problems involving finding an angle of a triangle .


2. 2. Solve Solve REAL LIFE problems problems that involve finding an that involve finding an angle of a triangle.angle of a triangle.

C

B

A18 Apr 202318 Apr 2023


a

b

c

The Cosine Rule can be used with ANY triangle as long as we have been given enough information.


cos2 2 2a =b +c - 2bc A

Finding Angles Finding Angles Using The Cosine RuleUsing The Cosine Rule

Consider the Cosine Rule again:We are going to change the subject of the formula to cos Ao

Turn the formula around:b2 + c2 – 2bc cos Ao = a2

Take b2 and c2 across.-2bc cos Ao = a2 – b2 – c2

Divide by – 2 bc.2 2 2

cos2

o a b cA

bc

Divide top and bottom by -12 2 2

cos2

o b c aA

bc

You now have a formula for finding an angle if you know all three sides of the triangle.


Write down the formula for cos Ao

2 2 2

cos2

o b c aA

bc

Label and identify Ao and a , b and c.

Ao = ? a = 11b = 9 c = 16

Substitute values into the formula.

2 2 29 16 11cos

2 9 16oA

Calculate cos Ao .Cos Ao =0.75

Use cos-1 0.75 to find Ao

Ao = 41.4o

Example 1 : Calculate the

unknown angle xo .



Example 2: Find the unknown

Angle in the triangle:

Write down the formula.

2 2 2

cos2

o b c aA

bc

Identify the sides and angle.

Ao = yo a = 26 b = 15 c = 13

2 2 215 13 26cos

2 15 13oA

Find the value of cosAo

cosAo = - 0.723The negative tells you the angle is obtuse.

Ao = yo =136.3o



What Goes In The Box ?What Goes In The Box ?

Calculate the unknown angles in the triangles below:

(1)

10m

7m5m Ao

Bo

(2) 12.7c

m

7.9cm

8.3cm

Ao =111.8o

Bo = 37.3o

18 Apr 202318 Apr 2023

Now try Ex 11.2Ch11 (page 143)

Cosine Rule Cosine Rule

18 Apr 202318 Apr 2023


2

1. A washing machine is reduced by 10%

in a sale. I t's sale price is £ 360.

What was the original price.

2. Factorise x - 7x +12

3. Find the missing angles. 61o

18 Apr 202318 Apr 2023


1.1. Be able to recognise the Be able to recognise the correct trigonometric correct trigonometric formula to use to solve a formula to use to solve a problem involving problem involving triangles.triangles.

1. To use our knowledge gained so far to solve various trigonometry problems.

Mixed problemsMixed 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

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

o 5020Cos

AC

o

5020

53.21 (2 )

ACCosm dp

53.21 m

o o

53.215 65

TCSin Sin

o

53.21 5 (1 )

655.1

SinTC m dp

Sin

B

T

C

180 – 110 = 70o 180 – 70 = 110o 180 – 115 = 65o

20o25o

5o

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)

H40 miles

24 miles

B

L

57 miles

A

2 2 257 40 242 57 40

CosAx x

A 20.4o

90 0 020.4 7 oBearing

2 2 2

2b c a

CosAbc

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.

2 2 2530 670 5202 530 670

CosPx x

48.7oP

180 22948.7 oBearing

P

670 miles

W

530 miles

Not to Scale

Q

520 miles

18 Apr 202318 Apr 2023

Now try Ex 14Ch8 (page 117)

Mixed Problems Mixed Problems

5-may-15 exact values angles greater than 90 o trigonometry useful notation & area of a triangle...

Documents

o x o

o slide

o sin

o sin

o cos

o cos

o tan

o tan