Trigonometric Functions - Contents

6. Trigonometric Results7. Trigonometric Graphs8. Graphical Solutions9. Derivatives of

Trigonometric Functions

10. Integrals of Trigonometric Functions

1. Radian Measure2. Area/

Circumference/Length of Arc

3. Area of a Sector4. Area of a minor

segment5. Small Angles

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Radian Measure

1 Uni

t

1 Unit

One radian is the angle that a one unit arc makes with centre of a unit circle.

1c

ProofCircumference =

2πr = 2π (1)= 2π

The circumference is 2π

2π = 360oπ = 180o

π radians = 180o

Radian Measure ConversionsExample 1

Convert into degrees.

5π3

Example 2

Convert 45o into radians.

Example 3

Convert 1.6c into degrees.

5x1803

5π3

=

= 300oπ = 180o 180o = π radians

1o = π/180 radians45o = π/180 x 45

radians= 45π/180 radians

= π/4 radiansπ radians = 180o

1 radians = 180/π deg

1.6 radians = 180/π x 1.6 deg

= 91o 40’

Area/Circumference/Length of Arc

r

Length of Arcl = r

( is in radians)

Area = r2

Circumference = 2r= D

Area/Circumference/Length of Arc

3 cmArea = r2

Circumference = 2r= 2 x 3

Length of Arcl = r

( is in radians)

4

= x 32

= 9 (exact)

≈ 28.3 cm2

(Approximate)

= 6 (exact)

≈ 18.8 cm(Approximate)

= 3 x /4= 3/4 (exact)

≈ 2.4 cm (Approximate)

= D

Area of a SectorArea of Sector

A = ½ r2

( is in radians)r

ProofArea of SectorArea of Circle

= Angle Revolution

Ar2

= 2

A = ½ r2

A = r22

ExampleFind the area of a sector with radius

3cm and angle /6.

A = ½ r2

= ½ 32

x /6.= 3/4 cm2.≈ 2. 36 cm2.

Area of a Minor SegmentArea of Sector

A = ½ r2 ( - sin )( is in radians) r

ProofArea of Minor

Segment

= Area ofSector

- Area ofTriangle

= ½ r2 - ½ r2 sin

Example Find the area of a minor segment formed with radius

3cm and angle /6.

= ½ 32 x (/6 - sin /6)

= (3/4 – 3/8)cm2 ≈ 1. 98 cm2.

= ½ r2 ( - sin )

= ½ r2 ( - sin

)

= ½ 32 x (/6 - ½)

b

a

Area = ½ab sin

Small Radian Angles

For small angles

Sin x ≈ xTan x ≈ xCos x ≈ 1

a=1x

h

o

Therefore

lim Sin x = 1xx0

lim Tan x = 1xx0

Cos x = a/hWhat happens to

a/h as x 0.a/h 1/1 1

Small Radian Angles

For small angles

Sin x ≈ xTan x ≈ xCos x ≈ 1

a=1x

h

o

Therefore

lim Sin x = 1xx0

lim Tan x = 1xx0

Sin x = o/hWhat happens to

o/h as x 0.o/h x 0/1 0

Small Radian Angles

For small angles

Sin x ≈ xTan x ≈ xCos x ≈ 1

a=1x

h

o

Therefore

lim Sin x = 1xx0

lim Tan x = 1xx0

Tan x = o/aWhat happens to

o/a as x 0.o/a 0/1 0

= 5

Small Radian Angles

Example:

Evaluate: lim Sin 5xxx0

lim 5(Sin 5x)5xx0x

lim Sin 5xx0 =

5 lim (Sin 5x)5xx0=

= 5 x 1

Trigonometric Results

AS

T C

0

π/2

π

Note:π=180o

θπ - θ

π + θ 2π - θ

3π/2

2π

1st 2nd

3rd 4th

Trigonometric Results

2

2

2

60o 60o

60o

π/3

π/3

π/3π

/6

1

√3

Sin π/3 = √3/2

Cos π/3 = 1/2

Sin π/6 = 1/2

Cos π/6 = √3/2

π/6 = 30oπ/3 = 60o

Tan π/6 = 1/√3Tan π/3 = √3

Trigonometric Results

1

1

√2

45o

45oπ/4

π/4

Sin π/4 = 1/√2

Cos π/4 = 1/√2

Tan π/4 = 1

π/4 = 45o

Trigonometric Results

Find the exact value of:

Sin =

5π4

5x180o

4

Sin = Sin 225o

= Sin (180o + 45o)= -Sin 45o

= -1√2

AS

CT

Sin is -ve in the 3rd Quadrant.

Trigonometric Results

Solve for 0 ≤ θ ≤ 2π

Evaluate Sin θ = 1/2θ = 30o

AS

CT

Sin is +ve in the 1st & 2nd Quadrants.

θ = π/6

θ = π - π/6

θ = 5π/6

30o

2

1

Trigonometric Graphs

y = sin(x)

π/23π/2

Geogebra

Trigonometric Graphs

y = cos(x)

π/23π/2

Geogebra

Trigonometric Graphs

y = tan(x)

π/23π/2

Geogebra

Trigonometric Graphs

y = cosec(x)

π/2 3π/2

Geogebra

Trigonometric Graphs

y = sec(x)

π/2 3π/2

Geogebra

Trigonometric Graphs

y = cot(x)

π/2 3π/2

Geogebra

y = 4 sin(x)

Trigonometric Graphs

Geogebra

y = sin(x)y = 2sin(x)y = 3sin(x)

The a Sin x Family of Curves

y = sin 2xy = sin 3xy = sin 4x

Trigonometric Graphs

The Sin bx Family of Curves

y = sin x

Geogebra

y = sin x - 1y = sin x

Trigonometric Graphs

The Sin x + c Family of Curves

Geogebra

y = sin x + 1

Graphical Solution

Graphical solve y = cos x and y = x

y = cos x

y = x

π/2π/4

One Solution

Derivative of Trigonometric Functions

ddx

(sin x) = cos xddx[sin f(x)] = f’(x) cos f(x)

ddx

(cos x) = sin x ddx

[cos f(x)] = -f’(x) sin f(x)

Sin x

Cos x

ddx

(tan x) = sec2 x ddx

[tan f(x)] = f’(x) sec2 f(x)

Derivative Function of Function Rule

Tan x

Derivative of Trigonometric Functions

ddx

(sin 4x) = 4 cos 4x

Sin 4x

ddx[tan (3x + 1)] = 3 sec2 (3x + 1)

Derivative Examples

Tan (3x + 1)

ddx[cos (x2 - 2x + 1)] = -(2x - 2) sin (x2 -2x + 1)

Cos (x2 - 2x + 1)

= 2(1 - x) sin (x2 -2x + 1)

ddx[sin f(x)] = f’(x) cos f(x)

ddx

[tan f(x)] = f’(x) sec2 f(x)

ddx

[cos f(x)] = -f’(x) sin f(x)

Integrals of Trigonometric Functions

Sin x

Cos x

Integral Function of Function Rule

Tan x

sin x dx = - cos x + C∫

cos x dx = sin x + C∫

sec2 x dx = tan x + c∫

∫sin (ax +b) dx = - 1/a cos (ax +

b) + C

cos (ax +b) dx = 1/a sin (ax + b)

+ C

∫

sec2 (ax +b) dx = 1/a tan (ax + b)

+ C

∫

Integral of Trigonometric Functions

Integration Examples

∫sin (ax +b) dx = - 1/a cos (ax + b)

+ C

∫sin (3x +2) dx = -1/3 cos (3x + 2)

+ C

Integral of Trigonometric Functions

Integration Examples

sin x dx = - cos x + C∫

∫sin x dx =0

π2

- cos x0

π2

= - cos - (-cos 0)

π2

= 0 – (-1)= 1