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Transcript of Trigonometry and Inverse Trigonometry - Emathsemaths.co.uk/images/stories/zoo/uploads/ebook math...

Trigonometry and Inverse Trigonometry

Maths Extension Trigonometry

Trigonometry

Trigonometric Ratios

Exact Values & Triangles

Trigonometric Identities

All Silver Tea Cups rule

Trigonometric Graphs

Sine & Cosine Rules

Area of a Triangle

Trigonometric Equations

Sums and Differences of angles

Double Angles

Triple Angles

Half Angles

T formula

Subsidiary Angle formula

General Solutions of Trigonometric Equations

Radians

Arcs, Sectors, Segments

Trigonometric Limits

Differentiation of Trigonometric Functions

Integration of Trigonometric Functions

Integration of sin2x and cos2x

INVERSE TRIGNOMETRY

Inverse Sin Graph, Domain, Range, Properties

Inverse Cos Graph, Domain, Range, Properties

Inverse Tan Graph, Domain, Range, Properties

Differentiation of Inverse Trigonometric Functions

Integration of Inverse Trigonometric Functions

Trigonometric Ratios

3

2

360

0

a

b

c

a

b

c

a

b

r

Sine

sin

q

=

hypotenuse

opposite

Cosine

cos

q

=

hypotenuse

adjacent

Tangent

tan

q

=

adjacent

opposite

Cosecant

cosec

q

=

q

sin

1

=

opposite

hypotenuse

Secant

sec

q

=

q

cos

1

=

adjacent

hypotenuse

Cotangent

cot

q

=

q

tan

1

=

opposite

adjacent

sin

q

=

(

)

q

-

90

cos

cos

q

=

(

)

q

-

90

sin

tan

q

=

(

)

q

-

90

cot

cosec

q

=

(

)

q

-

90

sec

sec

q

=

(

)

q

-

90

cos

ec

cot

q

=

(

)

q

-

90

tan

60 seconds = 1 minute60 = 1

60 minutes = 1 degree

60 = 1

q

q

q

cos

sin

tan

=

q

q

q

sin

cos

cot

=

Exact Values & Triangles

l

r

0

30

60

45

90

180

sin

0

2

1

2

3

2

1

1

0

cos

1

2

3

2

1

2

1

0

1

tan

0

3

1

3

1

0

cos

ec

2

3

2

2

1

sec

1

3

2

2

2

1

cot

3

3

1

1

0

Trigonometric Identities

q

q

2

2

cos

sin

+

= 1

q

2

cos

=

q

2

sin

1

-

q

2

sin

=

q

2

cos

1

-

q

2

cot

1

+

= cosec2

q

q

2

cot

= cosec2

q

1

1

= cosec2

q

q

2

cot

1

tan

2

+

q

=

q

2

sec

q

2

tan

=

1

sec

2

-

q

1

=

q

q

2

2

tan

sec

-

All Silver Tea Cups Rule

r

First Quadrant: All positive

q

sin

q

sin

+

q

cos

q

cos

+

q

tan

q

tan

+

Second Quadrant: Sine positive

(

)

q

-

180

sin

q

sin

+

(

)

q

-

180

cos

q

cos

(

)

q

-

180

tan

q

tan

Third Quadrant: Tangent positive

(

)

q

+

180

sin

q

sin

(

)

q

+

180

cos

q

cos

(

)

q

+

180

tan

q

tan

+

Fourth Quadrant: Cosine positive

(

)

q

-

360

sin

q

sin

(

)

q

-

360

cos

q

cos

+

(

)

q

-

360

tan

q

tan

2

p

-

Trigonometric Graphs

Sine & Cosine Rules

Sine Rule:

EMBED Equation.3

C

c

B

b

A

a

sin

sin

sin

=

=

OR

c

C

b

B

a

A

sin

sin

sin

=

=

2

p

Cosine Rule:

A

bc

c

b

a

cos

2

2

2

2

-

+

=

2

p

Area of a Triangle

C

ab

A

sin

2

1

=

C is the angle

a

&

b

are the two adjacent sides

p

2

p

-

2

p

Trigonometric Equations

Check the domain eg.

360

0

q

Check degrees (

360

0

q

) or radians (

p

q

2

0

)

If double angle, go 2 revolutions

If triple angle, go 3 revolutions, etc

If half angles, go half or one revolution (safe side)

Example 1

Solve sin =

2

1

for

360

0

q

q

sin

=

2

1

q

= 30, 150

Example 2

Solve cos 2 =

2

1

for

360

0

q

q

2

cos

=

2

1

q

2

= 60, 300, 420, 660

q

= 30, 150, 210, 330

Example 3

Solve tan

2

q

= 1 for

360

0

q

tan

2

q

= 1

2

q

= 45, 225

q

= 90

Example 4

0

cos

2

sin

=

+

q

q

q

q

q

cos

cos

sin

2

+

= 0

(

)

1

sin

2

cos

+

q

q

= 0

q

cos

= 0

q

sin

=

2

1

-

q

= 90, 270

q

= 210, 330

Example 5

2

2

cos

sin

3

-

=

-

q

q

(

)

q

q

2

sin

2

1

sin

3

-

-

= 2

1

sin

3

sin

2

2

+

+

q

q

= 0

(

)

(

)

1

sin

1

sin

2

+

+

q

q

= 0

q

sin

=

2

1

-

q

sin

= 1

q

= 210, 330

q

= 270

Sums and Differences of angles

(

)

b

a

+

sin

=

b

a

b

a

sin

cos

cos

sin

+

(

)

b

a

-

sin

=

b

a

b

a

sin

cos

cos

sin

-

(

)

b

a

+

cos

=

b

a

b

a

sin

sin

cos

cos

-

(

)

b

a

-

cos

=

b

a

b

a

sin

sin

cos

cos

+

(

)

b

a

+

tan

=

b

a

b

a

tan

tan

1

tan

tan

-

+

(

)

b

a

-

tan

=

b

a

b

a

tan

tan

1

tan

tan

+

-

Double Angles

q

2

sin

=

q

q

cos

sin

2

q

2

cos

=

q

q

2

2

sin

cos

-

=

q

2

sin

2

1

-

=

1

cos

2

2

-

q

q

2

tan

=

q

q

2

tan

2

1

tan

2

-

q

2

sin

=

(

)

q

2

cos

1

2

1

-

q

2

cos

=

(

)

q

2

cos

1

2

1

+

Triple Angles

q

3

sin

=

q

q

3

sin

4

sin

3

-

q

3

cos

=

q

q

cos

3

cos

4

3

-

q

3

tan

=

q

q

q

2

3

tan

3

1

tan

tan

3

-

-

Half Angles

q

sin

=

2

2

cos

sin

2

q

q

q

cos

=

2

2

2

2

sin

cos

q

q

-

=

2

2

sin

2

1

q

-

=

1

cos

2

2

2

-

q

q

tan

=

2

2

2

tan

2

1

tan

2

q

q

-

Deriving the Triple Angles

q

3

sin

=

(

)

q

q

+

2

sin

=

q

q

q

q

sin

2

cos

cos

2

sin

+

=

(

)

q

q

q

q

q

sin

sin

2

1

cos

cos

sin

2

2

-

+

=

q

q

q

q

3

2

sin

2

sin

cos

sin

2

-

+

=

(

)

q

q

q

q

3

2

sin

2

sin

sin

1

sin

2

-

+

-

=

q

q

q

q

3

3

sin

2

sin

sin

2

sin

2

-

+

-

=

q

q

3

sin

4

sin

3

-

_

Normal double angle_

Expand double angle_

Multiply_

Change

1

cos

sin

2

2

=

+

q

q

_

Simplify_

q

3

cos

=

(

)

q

q

+

2

cos

=

q

q

q

q

sin

2

sin

cos

2

cos

-

=

(

)

q

q

q

q

q

sin

cos

sin

2

cos

1

cos

2

2

-

-

=

q

q

q

q

cos

sin

2

cos

cos

2

2

3

-

-

=

(

)

q

q

q

q

cos

cos

1

2

cos

cos

2

2

3

-

-

-

=

q

q

q

q

3

2

cos

2

cos

2

cos

cos

2

+

-

-

=

q

q

cos

3

cos

4

3

-

q

3

tan

=

(

)

q

q

+

2

tan

=

q

q

q

q

tan

2

tan

1

tan

2

tan

-

+

=

q

q

q

q

q

q

2

2

tan

1

tan

tan

2

tan

1

tan

2

1

tan

-

-

-

+

=

q

q

q

q

q

q

q

2

2

2

2

3

tan

1

tan

2

tan

1

tan

1

tan

tan

tan

2

-

-

-

-

-

+

=

q

q

q

2

3

tan

3

1

tan

tan

3

-

-

T Formulae

Let t = tan

2

q

q

sin

=

2

1

2

t

t

+

q

cos

=

2

2

1

1

t

t

+

-

q

tan

=

2

1

2

t

t

-

q

sin

=

2

2

cos

sin

2

q

q

=

2

2

2

2

2

2

sin

cos

cos

sin

2

q

q

q

q

+

=

2

2

2

2

2

2

2

2

2

2

cos

sin

cos

cos

cos

sin

2

q

q

q

q

q

q

+

=

2

2

2

tan

1

tan

2

q

q

+

=

2

1

2

t

t

+

Using half angles

_

Divide by 1

1

cos

sin

2

2

=

+

q

q

Divide top and bottom by

q

2

cos

cos

cancel;

cos

sin

becomes tan

q

cos

=

2

2

2

2

sin

cos

q

q

-

=

2

2

2

2

2

2

2

2

sin

cos

sin

cos

q

q

q

q

+

-

=

2

2

2

2

2

2

2

2

2

2

2

2

cos

sin

cos

cos

sin

cos

q

q

q

q

q

q

+

-

=

2

2

2

2

tan

1

tan

1

q

q

+

-

=

2

2

1

1

t

t

+

-

q

tan

=

q

q

cos

sin

=

2

2

2

1

1

1

2

t

t

t

t

+

-

+

=

2

1

2

t

t

-

Subsidiary Angle Formula

x

b

x

a

cos

sin

+

=

)

sin

cos

cos

(sin

x

x

x

x

R

+

=

x

x

R

x

x

R

sin

cos

cos

sin

+

a

=

x

R

cos

2

a

\

=

x

R

2

2

cos

b

=

x

R

sin

2

b

\

=

x

R

2

2

sin

1

cos

sin

2

2

=

+

x

x

=

2

2

2

R

b

a

+

2

2

b

a

R

+

=

a

b

=

a

tan

x

b

x

a

cos

sin

+

= C

)

sin(

a

+

x

R

x

b

x

a

cos

sin

-

= C

)

sin(

a

-

x

R

x

b

x

a

sin

cos

+

= C

)

cos(

a

+

x

R

x

b

x

a

sin

cos

-

= C

)

cos(

a

-

x

R

Example 1

Find x.

1

cos

sin

3

=

-

x

x

R

=

2

2

1

3

+

a

tan

=

3

1

=

4

= 2

a

= 30

)

30

sin(

2

-

x

)

30

sin(

-

x

30

-

x

x

= 1

=

2

1

= 30, 150

= 60, 180

General Solutions of Trigonometric Equations

a

q

sin

sin

=

Then

a

p

q

n

n

)

1

(

-

+

=

a

q

cos

cos

=

Then

a

p

q

=

n

2

a

q

tan

tan

=

Then

a

p

q

+

=

n

Radians

c

p

= 180

1

=

180

c

p

Arcs, Sectors, Segments

Arc Length

1-1

x

y

2-2

x

y

l

=

q

r

Area of Sector

2

-2

x

y

A

=

q

2

2

1

r

Area of Segment

A

=

(

)

q

q

sin

2

2

1

-

r

Trigonometric Limits

x

x

x

sin

lim

0

=

x

x

x

tan

lim

0

=

x

x

cos

lim

0

= 1

Differentiation of Trigonometric Functions

(

)

x

dx

d

sin

=

x

cos

[

]

)

(

sin

x

f

dx

d

=

)

(

cos

)

(

'

x

f

x

f

(

)

)

sin(

b

ax

dx

d

+

=

)

cos(

b

ax

a

+

(

)

x

dx

d

cos

=

x

sin

-

[

]

)

(

cos

x

f

dx

d

=

)

(

sin

)

(

'

x

f

x

f

-

(

)

)

cos(

b

ax

dx

d

+

=

)

sin(

b

ax

a

+

-

(

)

x

dx

d

tan

=

x

2

sec

[

]

)

(

tan

x

f

dx

d

=

)

(

sec

)

(

'

2

x

f

x

f

(

)

)

tan(

b

ax

dx

d

+

=

)

(

sec

2

b

ax

a

+

x

dx

d

sec

=

x

x

tan

.

sec

ecx

dx

d

cos

=

ecx

x

cos

.

cot

-

x

dx

d

cot

=

x

ec

2

cos

-

Integration of Trigonometric Functions

ax

cos

dx

=

c

ax

a

+

sin

1

ax

sin

dx

=

c

ax

a

+

-

cos

1

ax

2

sec

dx

=

c

ax

a

+

tan

1

-

2

2

1

x

a

dx

=

c

a

x

+

-

1

sin

-

-

2

2

1

x

a

dx

=

c

a

x

+

-

1

cos

__OR__

c

a

x

+

-

-

1

sin

+

2

2

1

x

a

dx

=

c

a

x

a

+

-

1

tan

1

ax

ec

2

cos

dx

=

c

ax

a

+

-

cot

1

ax

ax

tan

.

sec

dx

=

c

ax

a

+

sec

1

ax

ecax

cot

.

cos

dx

=

c

ecax

a

+

-

cos

1

Integration of sin2x and cos2x

x

2

cos

1

2

cos

+

x

(

)

1

2

cos

2

1

+

x

=

1

cos

2

2

-

x

=

x

2

cos

2

=

x

2

cos

x

2

cos

dx

=

(

)

+

1

2

cos

2

1

x

dx

=

(

)

C

x

x

+

+

2

sin

2

1

2

1

=

C

x

x

+

+

2

1

4

1

2

sin

x

2

cos

dx

=

C

x

x

+

+

2

1

4

1

2

sin

x

2

cos

x

2

sin

2

x

2

sin

=

x

2

sin

1

-

=

x

2

cos

1

-

=

(

)

x

2

cos

1

2

1

-

x

2

sin

dx

=

(

)

-

x

2

cos

1

2

1

dx

=

(

)

C

x

x

+

-

2

sin

2

1

2

1

=

C

x

x

+

-

2

sin

4

1

2

1

x

2

sin

dx

=

C

x

x

+

-

2

sin

4

1

2

1

INVERSE TRIGNOMETRY

Inverse Sin Graph, Domain, Range, Properties

1

1

-

x

2

2

p

p

-

y

x

x

1

1

sin

)

(

sin

-

-

-

=

-

Inverse Cos Graph, Domain, Range, Properties

1

1

-

x

p

y

0

x

x

1

1

cos

)

(

cos

-

-

-

=

-

p

Inverse Tan Graph, Domain, Range, Properties

All real x

2

2

p

p

-

y

x

x

1

1

tan

)

(

tan

-

-

-

=

-

Differentiation of Inverse Trigonometric Functions

(

)

x

dx

d

1

sin

-

=

2

1

1

x

-

(

)

a

x

dx

d

1

sin

-

=

2

2

1

x

a

-

(

)

)

(

sin

1

x

f

dx

d

-

=

2

)]

(

[

1

)

(

'

x

f

x

f

-

(

)

x

dx

d

1

cos

-

=

2

1

1

x

-

-

(

)

a

x

dx

d

1

cos

-

=

2

2

1

x

a

-

-

(

)

)

(

cos

1

x

f

dx

d

-

=

2

)]

(

[

1

)

(

'

x

f

x

f

-

-

(

)

x

dx

d

1

tan

-

=

2

1

1

x

+

(

)

a

x

dx

d

1

tan

-

=

2

2

x

a

a

+

(

)

)

(

tan

1

x

f

dx

d

-

=

2

)]

(

[

)

(

'

x

f

a

x

f

+

Integration of Inverse Trigonometric Functions

-

2

2

1

x

a

dx

=

c

a

x

+

-

1

sin

-

-

2

2

1

x

a

dx

=

c

a

x

+

-

1

cos

__OR__

c

a

x

+

-

-

1

sin

+

2

2

1

x

a

dx

=

c

a

x

a

+

-

1

tan

1

opposite

adjacent

adjacent

opposite

hypotenuse

hypotenuse

EMBED Equation.3

1

2

30

60

1

1

EMBED Equation.3

45

EMBED Equation.3

90

180

270

SA

TC

1st Quadrant

4th Quadrant

2nd Quadrant

3rd Quadrant

A

B

C

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

A

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

C

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

Segment

EMBED Equation.3

EMBED Equation.3

0

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

EMBED Equation.3

Addu High School/Department of Mathematics/Resources/Math-ebook_Trigonometry

Page 2 of 23

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