Trigonometry and Inverse 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
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|Sine |sin[pic] | |= [pic] | |
|Cosine |cos[pic] | |= [pic] | |
|Tangent |tan[pic] | |= [pic] | |
|Cosecant |cosec[pic] |= [pic] |= [pic] | |
|Secant |sec[pic] |= [pic] |= [pic] | |
|Cotangent |cot[pic] |= [pic] |= [pic] | |
|sin[pic] |= [pic] |
|cos[pic] |= [pic] |
|tan[pic] |= [pic] |
|cosec[pic] |= [pic] |
|sec[pic] |= [pic] |
|cot[pic] |= [pic] |
60 seconds = 1 minute 60’’ = 1’
60 minutes = 1 degree 60’ = 1°
|[pic] |[pic] |
Exact Values & Triangles
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| |0° |30° |60° |45° |90° |180° |
|[pic] |0 |[pic] |[pic] |[pic] |1 |0 |
|[pic] |1 |[pic] |[pic] |[pic] |0 |–1 |
|[pic] |0 |[pic] |[pic] |1 |–– |0 |
|[pic]ec |–– |2 |[pic] |[pic] |1 |–– |
|[pic] |1 |[pic] |2 |[pic] |–– |–1 |
|[pic] |–– |[pic] |[pic] |1 |0 |–– |
Trigonometric Identities
|[pic] |= 1 |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= cosec2[pic] |
|[pic] |= cosec2[pic] – 1 |
|1 |= cosec2[pic] – [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|1 |= [pic] |
All Silver Tea Cups Rule
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|First Quadrant: All positive |
|[pic] |[pic] |+ |
|[pic] |[pic] |+ |
|[pic] |[pic] |+ |
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|Second Quadrant: Sine positive |
|[pic] |[pic] |+ |
|[pic] |–[pic] |– |
|[pic] |–[pic] |– |
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|Third Quadrant: Tangent positive |
|[pic] |–[pic] |– |
|[pic] |–[pic] |– |
|[pic] |[pic] |+ |
| |
|Fourth Quadrant: Cosine positive |
|[pic] |–[pic] |– |
|[pic] |[pic] |+ |
|[pic] |–[pic] |– |
Trigonometric Graphs
Sine & Cosine Rules
Sine Rule:
[pic][pic] OR [pic]
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Cosine Rule:
[pic]
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Area of a Triangle
|[pic] |C is the angle |
| |[pic] & [pic] are the two adjacent sides |
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Trigonometric Equations
➢ Check the domain eg. [pic]
➢ Check degrees ([pic]) or radians ([pic])
➢ 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 = [pic] for [pic]
|[pic] |= [pic] |
|[pic] |= 30°, 150° |
Example 2
Solve cos 2θ = [pic] for [pic]
|[pic] |= [pic] |
|[pic] |= 60°, 300°, 420°, 660° |
|[pic] |= 30°, 150°, 210°, 330° |
Example 3
Solve tan[pic] = 1 for [pic]
|tan[pic] |= 1 |
|[pic] |= 45°, 225° |
|[pic] |= 90° |
Example 4
[pic]
|[pic] |= 0 |
|[pic] |= 0 |
|[pic] |= 0 |[pic] |= [pic] |
|[pic] |= 90°, 270° |[pic] |= 210°, 330° |
Example 5
[pic]
|[pic] |= –2 |
|[pic] |= 0 |
|[pic] |= 0 |
|[pic] |= [pic] |[pic] |= –1 |
|[pic] |= 210°, 330° |[pic] |= 270° |
Sums and Differences of angles
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
Double Angles
|[pic] |= [pic] |
|[pic] |= [pic] |
| |= [pic] |
| |= [pic] |
|[pic] |= [pic] |
| | |
|[pic] |= [pic] |
|[pic] |= [pic] |
Triple Angles
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
Half Angles
|[pic] |= [pic] |
|[pic] |= [pic] |
| |= [pic] |
| |= [pic] |
|[pic] |= [pic] |
| | |
Deriving the Triple Angles
|[pic] |= [pic] |_ |
| |= [pic] |Normal double angle_ |
| |= [pic] |Expand double angle_ |
| |= [pic] |Multiply_ |
| |= [pic] |Change [pic]_ |
| |= [pic] |Simplify_ |
| |= [pic] | |
| | | |
|[pic] |= [pic] | |
| |= [pic] | |
| |= [pic] | |
| |= [pic] | |
| |= [pic] | |
| |= [pic] | |
| |= [pic] | |
| | | |
|[pic] |= [pic] | |
| |= [pic] | |
| |= [pic] | |
| |= [pic] | |
| |= [pic] | |
T – Formulae
Let t = tan[pic]
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |Using half angles |
| |= [pic] |_ |
| | |Divide by “1” |
| |= [pic] |[pic] |
| | | |
| |= [pic] | |
| |= [pic] | |
| | |Divide top and bottom by [pic] |
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| | |[pic]’ cancel; [pic] becomes tan |
|[pic] |= [pic] |[pic] |= [pic] |
| |= [pic] | | |
| | | |= [pic] |
| |= [pic] | | |
| | | |= [pic] |
| |= [pic] | | |
| |= [pic] | | |
Subsidiary Angle Formula
|[pic] |= [pic] |
| |= [pic] |
|[pic] |= [pic] |[pic] |= [pic] |
|[pic] |= [pic] |[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] | |
| |[pic] |
|[pic] |= C |[pic] |
|[pic] |= C |[pic] |
|[pic] |= C |[pic] |
|[pic] |= C |[pic] |
Example 1
Find x. [pic]
|R |= [pic] |[pic] |= [pic] |
| |= [pic] |[pic] |= 30° |
| |= 2 | | |
|[pic] |= 1 |
|[pic] |= [pic] |
|[pic] |= 30°, 150° |
|[pic] |= 60°, 180° |
General Solutions of Trigonometric Equations
|[pic] |Then [pic] |
|[pic] |Then [pic] |
|[pic] |Then [pic] |
Radians
|[pic] |= 180° |
|1° |= [pic] |
Arcs, Sectors, Segments
|Arc Length |
|[pic] |= [pic] | |
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|Area of Sector |
|A |= [pic] | |
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|Area of Segment |
|A |= [pic] | |
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Trigonometric Limits
|[pic] |= [pic] |= [pic] |= 1 |
Differentiation of Trigonometric Functions
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
Integration of Trigonometric Functions
| |= [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
| |= [pic]__OR__ [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
Integration of sin2x and cos2x
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] dx |= [pic] dx |
| |= [pic] |
| |= [pic] |
|[pic] dx |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] dx |= [pic] dx |
| |= [pic] |
| |= [pic] |
|[pic] dx |= [pic] |
INVERSE TRIGNOMETRY
Inverse Sin – Graph, Domain, Range, Properties
| |[pic] |
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| |[pic] |
| |[pic] |
Inverse Cos – Graph, Domain, Range, Properties
| |[pic] |
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| |[pic] |
| |[pic] |
Inverse Tan – Graph, Domain, Range, Properties
| |All real x |
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| |[pic] |
| |[pic] |
Differentiation of Inverse Trigonometric Functions
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
|[pic] |= [pic] |
Integration of Inverse Trigonometric Functions
| |= [pic] |
|[pic] dx | |
| |= [pic]__OR__ [pic] |
|[pic] dx | |
| |= [pic] |
|[pic] dx | |
-----------------------
opposite
adjacent
adjacent
opposite
hypotenuse
hypotenuse
¸
¸
[pic]
1
2
30°
60°
1
1
[pic]
45°
[pic]
90°
180°
270°
S A
T C
1st Quadrant
4th Quadrant
2nd Quadrant
3rd Quadrant
A
B
C
[pic]
1
2
30°
60°
1
1
[pic]
45°
[pic]
90°
180°
270°
S A
T C
1st Quadrant
4th Quadrant
2nd Quadrant
3rd Quadrant
A
B
C
[pic]
[pic]
[pic]
A
[pic]
[pic]
[pic]
[pic]
[pic]
C
θ
[pic]
[pic]
θ
[pic]
θ
[pic]
Segment
[pic]
[pic]
0
[pic]
[pic]
[pic]
[pic]
................
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