Trigonometry and Inverse Trigonometry - Emathsemaths.co.uk/images/stories/zoo/uploads/ebook math...
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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 sin 2 x and cos 2 x 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 Addu High School/Department of Mathematics/Resources/Math-ebook_Trigonometry Page 1 of 29
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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
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 1 of 23
Maths Extension – 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°
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 2 of 23
θ
θhypotenuse hypotenuse
opposite
adja
cent
adjacent
oppo
site
Maths Extension – 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 =
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 3 of 23
1
1
2
45°
3
1
2
30°
60°
Maths Extension – Trigonometry
All Silver Tea Cups Rule
First Quadrant: All positive+++
Second Quadrant: Sine positive+
– –– –
Third Quadrant: Tangent positive– –– –
+
Fourth Quadrant: Cosine positive– –
+– –
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 4 of 23
3600
90°
180°
270°
S A
T C
1st Quadrant
4th Quadrant
2nd Quadrant
3rd Quadrant
Maths Extension – Trigonometry
Trigonometric Graphs
Sine & Cosine RulesSine Rule:
OR
Cosine Rule:
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 5 of 23
A
BC
a
b c
Maths Extension – Trigonometry
Area of a Triangle
C is the angle & are the two adjacent sides
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 6 of 23
C
A
a
b c
Maths Extension – 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
= –2Addu High School/Department of Mathematics/Resources/Math-ebook_Trigonometry
Page 7 of 23
Maths Extension – Trigonometry
= 0= 0
= = –1= 210°, 330°
= 270°
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 8 of 23
Maths Extension – Trigonometry
Sums and Differences of angles
= = = = =
=
Double Angles=
= = =
=
= =
Triple Angles= = =
Half Angles=
= = =
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 9 of 23
Maths Extension – Trigonometry
=
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 10 of 23
Maths Extension – Trigonometry
Deriving the Triple Angles
= = = = = = =
_Normal double angle_Expand double angle_Multiply_Change _Simplify_
= = = = = = =
= =
=
=
=
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 11 of 23
Maths Extension – Trigonometry
T – FormulaeLet t = tan
=
=
=
=
=
=
=
=
Using half angles_Divide by “1”
Divide top and bottom by
’ cancel; becomes tan
=
=
=
=
=
=
=
=
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 12 of 23
Maths Extension – Trigonometry
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 13 of 23
Maths Extension – Trigonometry
Subsidiary Angle Formula
= =
= = = =
=
= C= C= C= C
Example 1Find x.
R = = = = 2 = 30°
= 1= = 30°, 150°= 60°, 180°
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 14 of 23
Maths Extension – Trigonometry
General Solutions of Trigonometric Equations
Then
Then
Then
Radians= 180°
1° =
Arcs, Sectors, SegmentsArc Length
=
Area of SectorA =
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 15 of 23
θ
θ r
Maths Extension – Trigonometry
Area of SegmentA =
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 16 of 23
θ rSegment
Maths Extension – Trigonometry
Trigonometric Limits
= = = 1
Differentiation of Trigonometric Functions
=
=
=
=
=
=
=
=
=
=
=
=
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 17 of 23
Maths Extension – Trigonometry
Integration of Trigonometric Functions
dx =
dx =
dx =
dx =
dx = __OR__
dx =
dx =
dx =
dx =
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 18 of 23
Maths Extension – Trigonometry
Integration of sin 2 x and cos 2 x= = =
dx = dx= =
dx =
= = =
dx = dx= =
dx =
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 19 of 23
Maths Extension – Trigonometry
INVERSE TRIGNOMETRYInverse Sin – Graph, Domain, Range, Properties
Inverse Cos – Graph, Domain, Range, Properties
Inverse Tan – Graph, Domain, Range, Properties
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 20 of 23
1-1x
y
0
2-2x
y
Maths Extension – Trigonometry
All real x
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 21 of 23
2
-2
x
y
Maths Extension – Trigonometry
Differentiation of Inverse Trigonometric Functions
=
=
=
=
=
=
=
=
=
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 22 of 23
Maths Extension – Trigonometry
Integration of Inverse Trigonometric Functions
dx =
dx = __OR__
dx =
Addu High School/Department of Mathematics/Resources/Math-ebook_TrigonometryPage 23 of 23
