Maths Extension 1 ± Trigonometry

Maths Extension 1 – Trigonometry

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

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Trigonometric Ratios

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°


θ

θ

hypotenuse hypotenuse

opposite

adja

cent

adjacent

oppo

site



Exact Values & Triangles

0° 30° 60° 45° 90° 180°0 1 01 0 –10 1 –– 0

ec –– 2 1 ––1 2 –– –1–– 1 0 ––

Trigonometric Identities

= 1= =

= cosec2

= cosec2 – 1 1 = cosec2 –

= =

1 =


1

1

2

45°

3

1

2

30°

60°



ASTC Rule

First Quadrant: All positive+++

Second Quadrant: Sine positive+

– –– –

Third Quadrant: Tangent positive– –– –

+

Fourth Quadrant: Cosine positive– –

+– –


360

0

90°

180°

270°

S A

T C

1st Quadrant

4th Quadrant

2nd Quadrant

3rd Quadrant



Trigonometric Graphs

Sine & Cosine RulesSine Rule:

OR

Cosine Rule:


A

BC

a

b c

A

a

b c



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 1Solve sin θ = for

= = 30°, 150°

Example 2Solve cos 2θ = for

= = 60°, 300°, 420°, 660°= 30°, 150°, 210°, 330°

Example 3Solve tan = 1 for

tan = 1= 45°, 225°= 90°

Example 4

= 0= 0

= 0 = = 90°, 270°

= 210°, 330°

Example 5

= –2




= 0= 0

= = –1= 210°, 330°

= 270°




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 – FormulaeLet t = tan

=

=

=

=

=

=

=

=

Using half angles_Divide by “1”

Divide top and bottom by

’ cancel; becomes tan

=

=

=

=

=

=

=

=




Subsidiary Angle Formula

= =

= = = =

=

= C= C= C= C

Example 1Find x.

R = =

= = 2

= 30°

= 1= = 30°, 150°= 60°, 180°




General Solutions of Trigonometric Equations

Then

Then

Then

Radians

= 180°

1° =

Arcs, Sectors, SegmentsArc Length

=

Area of SectorA =


θ

θ r



Area of SegmentA =


θr

Segment



Trigonometric Limits

= = = 1

Differentiation of Trigonometric Functions

=

=

=

=

=

=

=

=

=

=

=

=




Integration of Trigonometric Functions

dx =

dx =

dx =

dx =

dx = __OR__

dx =

dx =

dx =

dx =




Integration of sin 2 x and cos 2 x= = =

dx = dx= =

dx =

= = =

dx = dx= =

dx =




INVERSE TRIGNOMETRYInverse Sin – Graph, Domain, Range, Properties

Inverse Cos – Graph, Domain, Range, Properties

Inverse Tan – Graph, Domain, Range, Properties

All real x


1-1

x

y

0

2-2

x

y

2

-2

x

y



Differentiation of Inverse Trigonometric Functions

=

=

=

=

=

=

=

=

=




Integration of Inverse Trigonometric Functions

dx =

dx = __OR__

dx =



Maths Extension 1 ± Trigonometry

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