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] |+ |

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|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] |

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| |= [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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