TRIGONOMETRY - LESSON 1 Introduction

TRIGONOMETRY - LESSON 1

Trigonometry : L1 - Introduction


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sinθ

cosθ

tanθ

00

0

1

0

12

1√2

2√3 1

√21

√31

300 450

12

√3

0

Undefined

900

2√3

600

1

0π6

π4

π3

π2T-ratio

Angle (θ)

Trigonometric functions of particular angles


TRIGONOMETRIC IDENTITIES

1. sin2θ + cos2θ = 1

2. sec2θ – tan2θ = 1

3. cosec2θ – cot2θ = 1


Prove the following : sec4 x – sec2 x = tan4 x + tan2 x

Example: 1


Prove the following : sec4 x – sec2 x = tan4 x + tan2 x

Example: 1


If sinθ + sin2θ = 1 Prove that 1+ =cos12θ 3 cos10 θ+ +3cos8 θ cos6 θExample: 2


If sinθ + sin2θ = 1 Prove that 1+ =cos12θ 3 cos10 θ+ +3cos8 θ cos6 θExample: 2


Eg 3. If x = secθ - tanθ, y = cosecθ + cotθ, then xy + 1 = ……...

A

B

D

C

x + y

x - y

y - x

-x - y


Solution:


Q.If x = secθ - tanθ, y = cosecθ + cotθ, then xy + 1 = ……...

A

B

D

C

x + y

x - y

y - x

-x - y


Measurement of an Angle


Degrees

1. Sexagesimal system (British System)


Definition



It is denoted by 10

If the central angle is divided into 360 equalparts, each part in it is called One degree.

Definition

Again if 10 is divided into 60 equal parts each part in it called One minute.

It is denoted as 1′. 10 = 60′

&

y′

x′ x

y

O


Definition

It is denoted as 1′′

If each 1′ is subdivided into 60 equal parts eachpart in it is called One second.

1′ = 60′′




Radians

2. Circular system (or) Radian Measure


Definition

One radian is denoted as 1c.

A radian is an angle subtended at the centre ofa circle by an arc

r

B

A

r

r1c

whose length is equal to the radius of the circle.


2. Circular system (or) Radian Measure

Definition

Angle subtended at the centre of a circle of radius r by an arc of length l is defined as θ = l/r radians.


Relation between radians and degrees

∴ 1c =1800

π

1c ≈ (57.272…)0

3600 = 2πc

1c =(180)0

22

7

=0

11

630Clearly 1c • 10

90

11

Remember

➢ To convert radians into degree multiply with

➢ To convert degrees into radians multiply with

π1800

1800

π


Express the following angles in degrees.

i)5π12

c

=

ii)7π12

c

–

iii)13

c

=

=

Example:1


i

Express the following angles in degrees.

i)5π12

c

= 750

ii)7π12

c

– 105 0–

iii)13

c

=

=

60π

0

Example:1

Answer

iv) 11/16


=

Express the following angles in radians.

i) 1200

=ii) 6000–

=iii) 1440–

Example


=

Express the following angles in radians.

i) 1200

=ii) 6000–

2π c

3

=iii) 1440–

–10π 3

c

4πc 5

–

Example: 2

Answer

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Examples On l = r θ


Example:3 A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope tight and describes 88m when it has traced out 720 at the center, find the length of the rope.


Ar720

P

B

A horse is tied to a post by a rope. If the horse moves along a circular path always keeping the rope tight and describes 88m when it has traced out 720 at the center, find the length of the rope.

B

A

Example: 3

Answer

=length of the rope 70 meters


Example: 4


Example: 4


Circular Functions


Let P be a point on the circle such that OP makes angle θ with the x axis in Anticlockwise direction i.e. ∠AOP = θ

Consider a unit circle (i.e. the circle with radius = 1) with centre at the origin O.

O 1

y′

xA

Then cosθ and sin θare defined ascosθ = x – coordinate of Psin θ = y – coordinate of P

y′

x′ x

y

O 1

P(cos θ, sin θ)

θ


Remark

1. θ is positive in anticlockwise direction.

2. θ is negative in clockwise direction.

3. cosθ and sinθ are defined for all θ∈R. ( R : set of all real numbers)

x

y

θ

P(x , y)

OA(1,0)(-1,0)A’

B(0,1)

B’(0,-1)

-θ

4. -1≤ cosθ ≤ 1 -1 ≤ sin θ ≤ 1 for all θ∈R. ( R : set of all real numbers)


Sign of Trigonometric Functions


Sign of Trigonometric Functions

cosθ < 0 ; sinθ < 0

cosθ < 0 ; sinθ > 0 cosθ > 0 ; sinθ > 0

cosθ > 0 ; sinθ < 0

III

III IV

x

y

x > 0, y > 0x < 0, y > 0

x < 0, y < 0 x > 0, y < 0


y′

x′ x

y

O 1

P(cos θ, sin θ)

θ θ + 2π

P(cos (θ+ 2π), sin (θ+ 2π))

cosθ = cos (θ + 2π) = cos(θ + 4π)= …sinθ = sin (θ + 2π) = sin(θ + 4π)= …

In general


Other trigonometric ratios

1. tan θ = sin θ

cos θ π2

θ ≠ odd multiple of 2. sec θ =

1

cos θ

3. cot θ = cos θ

sin θ π,θ ≠ any multiple of 4. cosec θ =

1

sin θ


T-ratio

Angle (θ)

sinθ

cosθ

tanθ

900

(0, 1)

Not defined

1800 2700

0

3600

π 3π2

2π

Trigonometric functions of particular angles

π2

0

X′

Y

900

(cosθ, sinθ)

0

1 (–1, 0)

–1

0

(0, –1)0

–1

(1, 0)

1

0

1800

2700

3600

Not defined


1) Sin (765 )

2) Cos (1710)

3) Tan 19 (𝛑/3)

4) Sin 31(𝛑/3)





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Documents

TRIGONOMETRY - LESSON 1 Introduction