Higher Maths 2 3 Advanced Trigonometry1. Basic Trigonometric Identities 2Higher Maths 2 3 Advanced...

Higher Maths 2 3 Advanced Trigonometry

1

Basic Trigonometric Identities

2Higher Maths 2 3 Advanced Trigonometry

There are several basic trigonometric factsor identities which it is important to remember.

( sin x ) 2

is written

sin 2

x

sin 2

x + cos 2

x = 1

cos 2

x = 1 – sin 2

x

tan x =sin xcos x

Alternatively,

Example Find tan x if sin x

=cos

2 x = 1 – sin

2 x

= 1 –

= 59

cos x =

tan x =2

sin 2

x = 1 – cos 2

x 23 ÷ =3

55

35

49

23

Compound Angles


An angle which is the sum of two other angles is called a Compound Angle.

Angle SymbolsGreek letters are often used for angles.

‘Alpha

’‘Beta’

‘Theta

’‘Phi’

‘Lambd

a’

B

C

A

BAC =

BAC is acompound angle.

sin ( ) sin + sin

+

+ ≠

sin ( )Formula for

4

By extensive working,it is possible to prove that

+

sin ( )+ sin +sin≠

sin ( ) = sin cos + sin cos

+


Example

Find the exact value of sin 75°

sin 75° = sin ( 45° + 30° )

= sin 45°cos 30° + sin 30°cos 45°

= 2

312

× + × = 2 2+1

2132

1

Compound Angle Formulae

5

sin ( ) = sin cos + sin

cos +


sin ( ) = sin cos – sin

cos –

cos ( ) = cos cos – sin

sin+

cos ( ) = cos cos + sin

sin–

The result for sin ( )+ can be used to findall four basic compound angle

formulae.

Proving Trigonometric Identities


Example

Prove the identity

sin ( )+

cos costan + tan=

sin ( )+

cos cos=

cos cos

sin cos +sin cos

=cos cos

sin cos+

sin cos

cos cos

=cos

sin+

sin

cos= tan + tan

An algebraic fact is called an identity.

tan x sin xcos x=

‘Left Hand Side’

L.H.S.

R.H.S. ‘Right

Hand Side’

Applications of Trigonometric Addition Formulae


K

L

J

M

8

3 4

From the diagram, show

thatcos ( )– =

255

KL

= 82 + 42

80

= = 4 5

JK

= 32 + 42

25

= = 5Example

cos ( )– = cos cos + sin sin

=5

15

× + ×34 2

5 5

10 55

= = 25

255

=

cos =454

15

= sin =854

25

=

Find any unknown sides:

Investigating Double Angles


The sum of two identical angles can be written as and is called a double angle.

2

2sin = sin ( + ) = cos +sin cossin

= 2sin cos

2cos = cos ( + ) = cos –cos sinsin

= cos 2 – sin

2

= cos 2 – ( 1 – cos

2 )

= cos 2 – 12 or sin

2–1 2

sin 2

x + cos 2

x = 1

sin 2

x = 1 – cos 2

x

( )

Double Angle Formulae


There are several basic

identities for double

angles which it is

useful to know.

sin 2 = 2 sin cos

cos 2 = cos2 – sin2

= 2 cos2 – 1

= 1 – 2 sin2Example

3

4

If tan = ,

calculate

and .

34

5

sin 2 cos 2

sin 2 =2 sin cos

= 2 × 54 × 5

3

= 25

24

cos 2 = cos 2 – sin

2

= 53 –

= 25

7

2 ( )54 2

–

tan = adjopp

Trigonometric Equations involving Double Angles

10


cos 2 x – cos x = 0 Solve

for

0 x 2π

cos 2 x – cos x = 0

2 cos 2

x – 1 – cos x = 0 2 cos

2 x – cos x – 1 =

0 ( 2 cos x + 1) ( cos x – 1) = 0

cos x – 1 = 0

cos x = 1

x = 2π

2 cos x + 1 = 0

cos x = 21–S A

T

3π

x =C 3π4

3π2

or x = x = 0

or

or

substitu

te

remember

Example

Intersection of Trigonometric Graphs

11


4

-4

360°

A

B

f (x)g(x)

Example

The diagram opposite shows

the graphs of and

.

g(x)f (x)

Find the x-coordinate of A and B.

4 sin 2 x = 2 sin x

4 sin 2 x – 2 sin x = 0

4 × ( 2 sin x cos x ) – 2 sin x = 0

8 sin x cos x – 2 sin x = 0

2 sin x ( 4 cos x – 1 ) = 0 common factor

f (x) =

g(x) 2 sin x = 0

4 cos x – 1 = 0 or

x = 0°, 180° or 360°

orx ≈ 75.5° or 284.5°

Solving by trigonometry,

Quadratic Angle Formulae

12


The double angle formulae

can also be rearranged to

give quadratic angle

formulae.

cos2 = 21 ( 1 + cos 2

)sin2 = 2

1 ( 1 – cos 2

)Example

Express

in terms of cos 2 x .

2 cos 2

x – 3 sin 2

x2 × ( 1 + cos 2 x ) – 3 × ( 1 – cos

2 x ) 21

21

1 + cos 2 x – +

cos 2 x

23

23

25

21– + cos 2 x

=

=

= = 21( 5 cos 2 x – 1 )

substitute

Quadratic means

‘squared’

2 cos 2

x – 3 sin 2

x

Angles in Three Dimensions

13


In three dimensions, a flat

surface is called a plane.

Two planes at different

orientations have a straight

line of intersection.

A

B

C

D

P

Q

J

LK The angle between two

planes is defined as

perpendicular to the line of

intersection.

S

Three Dimensional Trigonometry

14


Challenge

P

Q

R

T

OS H×

÷ ÷ AC H×

÷ ÷ OT A×

÷ ÷

OM N8m 6m

Many problems in three dimensions

can be solved using Pythagoras and basic trigonometry

skills.

Find all unknown angles and

lengths in the pyramid shown

above.

9m

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