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Maths Extension 1 ± TrigonometryTrigonometry 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
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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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Maths Extension 1 – Trigonometry
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°
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θ
θ
hypotenuse hypotenuse
opposite
adja
cent
adjacent
oppo
site
Maths Extension 1 – Trigonometry
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 =
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1
1
2
45°
3
1
2
30°
60°
Maths Extension 1 – Trigonometry
ASTC Rule
First Quadrant: All positive+++
Second Quadrant: Sine positive+
– –– –
Third Quadrant: Tangent positive– –– –
+
Fourth Quadrant: Cosine positive– –
+– –
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360
0
90°
180°
270°
S A
T C
1st Quadrant
4th Quadrant
2nd Quadrant
3rd Quadrant
Maths Extension 1 – Trigonometry
Trigonometric Graphs
Sine & Cosine RulesSine Rule:
OR
Cosine Rule:
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A
BC
a
b c
A
a
b c
Maths Extension 1 – Trigonometry
Area of a Triangle
C is the angle & are the two adjacent sides
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C
Maths Extension 1 – Trigonometry
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
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Maths Extension 1 – Trigonometry
= 0= 0
= = –1= 210°, 330°
= 270°
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Maths Extension 1 – Trigonometry
Sums and Differences of angles
= = = =
=
=
Double Angles
=
= = =
=
= =
Triple Angles= =
=
Half Angles
=
= = =
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Maths Extension 1 – Trigonometry
=
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Maths Extension 1 – Trigonometry
Deriving the Triple Angles
= = = = = = =
_Normal double angle_Expand double angle_Multiply_Change _Simplify_
= = = = = = =
=
=
=
=
=
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Maths Extension 1 – Trigonometry
T – FormulaeLet t = tan
=
=
=
=
=
=
=
=
Using half angles_Divide by “1”
Divide top and bottom by
’ cancel; becomes tan
=
=
=
=
=
=
=
=
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Maths Extension 1 – Trigonometry
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Maths Extension 1 – Trigonometry
Subsidiary Angle Formula
= =
= = = =
=
= C= C= C= C
Example 1Find x.
R = =
= = 2
= 30°
= 1= = 30°, 150°= 60°, 180°
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Maths Extension 1 – Trigonometry
General Solutions of Trigonometric Equations
Then
Then
Then
Radians
= 180°
1° =
Arcs, Sectors, SegmentsArc Length
=
Area of SectorA =
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θ
θ r
Maths Extension 1 – Trigonometry
Area of SegmentA =
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θr
Segment
Maths Extension 1 – Trigonometry
Trigonometric Limits
= = = 1
Differentiation of Trigonometric Functions
=
=
=
=
=
=
=
=
=
=
=
=
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Maths Extension 1 – Trigonometry
Integration of Trigonometric Functions
dx =
dx =
dx =
dx =
dx = __OR__
dx =
dx =
dx =
dx =
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Maths Extension 1 – Trigonometry
Integration of sin 2 x and cos 2 x= = =
dx = dx= =
dx =
= = =
dx = dx= =
dx =
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Maths Extension 1 – Trigonometry
INVERSE TRIGNOMETRYInverse Sin – Graph, Domain, Range, Properties
Inverse Cos – Graph, Domain, Range, Properties
Inverse Tan – Graph, Domain, Range, Properties
All real x
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1-1
x
y
0
2-2
x
y
2
-2
x
y
Maths Extension 1 – Trigonometry
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Maths Extension 1 – Trigonometry
Differentiation of Inverse Trigonometric Functions
=
=
=
=
=
=
=
=
=
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Maths Extension 1 – Trigonometry
Integration of Inverse Trigonometric Functions
dx =
dx = __OR__
dx =
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