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Trigonometry - Maths Extension

Mathematic

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Trigonometry

Trigonometric Ratios

Exact Values & Triangles




Trigonometric Identities

ASTC 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

Text Box: opposite Text Box: adjacent


Sine

sin

=

Cosine

cos

=

Tangent

tan

=

Cosecant

cosec

=

=

Secant

sec

=

=

Cotangent

cot

=

=

sin

=

cos

=

tan

=

cosec

=

sec

=

cot

=

60 seconds = 1 minute 60’’ = 1’

60 minutes = 1 degree 60’ = 1°

Exact Values & Triangles


ec

Trigonometric Identities

=

=

= cosec2

= cosec2 – 1

= cosec2 –

=

=

=

ASTC Rule


First Quadrant: All positive

Second Quadrant: Sine positive

–

–

Third Quadrant: Tangent positive

–

–

Fourth Quadrant: Cosine positive

–

–


Trigonometric Graphs

Sine & Cosine Rules

Sine Rule:

OR


Cosine Rule:




Area of a Triangle

C is the angle

& are the two adjacent sides

 

 

C

 

Trigonometric Equations

Check the domain eg.

Check degrees () or radians ()

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 θ = for

=

Example 2

Solve cos 2θ = for

=

Example 3

Solve tan = 1 for

tan

Example 4

=

Example 5

=

Sums and Differences of angles

=

=

=

=

=

=

Double Angles

=

=

=

=

=

=

=

Triple Angles

=

=

=

Half Angles

=

=

=

=

=

Deriving the Triple Angles

=

=

=

=

=

=

=

Normal double angle_

Expand double angle_

Multiply_

Change _

Simplify_

=

=

=

=

=

=

=

=

=

=

=

=

T – Formulae

Let t = tan

=

=

=

=

=

=

=

=

Using half angles

Divide by “1”

Divide top and bottom by

’ cancel; becomes tan

=

=

=

=

=

=

=

=

Subsidiary Angle Formula

=

=

=

=

=

=

=

= C

= C

= C

= C

Example 1

Find x.

R

=

=

=



=

General Solutions of Trigonometric Equations

Then

Then

Then

Radians

=

Arcs, Sectors, Segments

Arc Length

 

 

=

Area of Sector

A

=

Area of Segment

A

=

Trigonometric Limits

=

=

Differentiation of Trigonometric Functions

=

=

=

=

=

=

=

=

=

=

=

=

Integration of Trigonometric Functions

dx

=

dx

=

dx

=

dx

=

dx

= __OR

dx

=

dx

=

dx

=

dx

=

Integration of sin2x and cos2x

=

=

=

dx

= dx

=

=

dx

=

=

=

=

dx

= dx

=

=

dx

=

INVERSE TRIGNOMETRY

Inverse Sin – Graph, Domain, Range, Properties


Inverse Cos – Graph, Domain, Range, Properties


Inverse Tan – Graph, Domain, Range, Properties


All real x

Differentiation of Inverse Trigonometric Functions

=

=

=

=

=

=

=

=

=

Integration of Inverse Trigonometric Functions

dx

=

dx

= __OR

dx

=






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