TRIGONOMETRIC FUNCTIONS OF ACUTE ANGLES

ByM. Jaya krishna Reddy

Mentor in mathematics,APIIIT-Basar, Adilabad(dt),A.P.

India.

Acute Angle:

ACUTE ANGLE

An angle whose measure is greater than zero but less than 90 is called an “acute angle”

o Initial ray

TERMINAL

RAY

sinopp b

hyp c

cosadj a

hyp c

tanopp b

adj a

coshyp c

ecopp b

sechyp c

adj a

cotadj a

opp b

Some Old Houses Can’t Always Hide

Their Old Age

Commonly used mnemonic for these ratios :

Ѳ

c

a

b

BC

A

•Trigonometric functions(also called circular functions) are functions of an angle.

History:

• They are used to relate the angles of a triangle to the lengths of the sides of a triangle.• Sumerian astronomers introduced angle measure, using a division of circles into 360 degrees.

• The sine function was first defined in the “surya siddhanta” and its properties were further documented by the fifth century Indian mathematician and astronomer “Aryabhatta”.

• By 10th century the six trigonometric functions were used.

Applications:

• In 240 B.C. a mathematician named “Eratosthenes” discovered the radius of the earth as 4212.48 miles using trigonometric functions..

• In 2001 a group of European astronomers did an experiment by using trigonometric functions and they got all the measurement, they calculate the Venus was about 105,000,000 km away from the sun and the earth was about 150, 000, 000 km away.

• Optics and statics are 2 early fields of Physics that use trigonometry.

• It is also the foundation of the practical art of surveying

sin cossin .cos

sec cosec

1. Prove that

sincos . tan cos . sin

cos

2. Prove that

sin cossin cos

sin .cos

sin cos sin cos1 1sec coscos sin

ec

cos . tan sin

Sol:

Sol:

sin .cos

Fundamental Relations:

sinopp b

hyp c

cosadj a

hyp c

Squaring and adding both the equations

2 22 2(sin ) (cos )

b a

c c

2 2 2

2 21

b a c

c c

2 2sin cos 1

Ѳ

c

a

b

C

A

B

From the above diagram, By

Pythagorean Rule, 2 2 2a b c

tanopp b

adj a

sechyp c

adj a

Squaring and subtracting the equations, we get

2 22 2(sec ) (tan )

c b

a a

2 2 2

2 21

c b a

a a

2 2sec tan 1

2 2sec cot 1co Similarly,

Ѳ

c

a

b

C

A

B

From the above diagram, By

Pythagorean Rule, 2 2 2a b c

2 2sin cos 1 2 2sec tan 1

2 2sec cot 1co

Example: Prove that2sec 1

sinsec

sol:

2tan tan

sec sec

2sec 1

secGiven that

sin cossin

cos 1

Ex: Prove that sec2Ѳ - cosec2Ѳ = tan2Ѳ - cot2Ѳ Sol: We know that sec2Ѳ - tan2Ѳ = 1 = cosec2Ѳ - cot2Ѳsec2Ѳ - tan2Ѳ = cosec2Ѳ - cot2Ѳ

sec2Ѳ - cosec2Ѳ = tan2Ѳ - cot2Ѳ

Example: Prove that 2 2

2 22 2

cos costan tan

cos .cos

B AA B

A B

Sol: Given that 2 2tan tanA B2 2

2 2

2 2

2 2 2 2

sec 1 (sec 1)

sec sec

1 1 cos cos

cos cos cos .cos

A B

A B

B A

A B A B

00 300 450 600 900

Sin 0 1

Cos 1 0

Tan 0 1 ∞

Cosec ∞ 2 1

Sec 1 2 ∞

Cot ∞ 1 0

1

2

1

23

2

3

2

1

21

21

33

2 2

32

3 2

31

3

Values of the trigonometrical ratios :

Example: find the value of tan450.sec300 - cot900.cosec450

Sol: Given that tan450.sec300 -- cot900.cosec450

= 1 . -- 0. =

2

32

2

3

Example: If cosѲ = 3/5, find the value of the other ratios

4sin

5

opp

hyp

4tan

3

opp

adj

5cos

4

hypec

opp

5sec

3

hyp

adj

3cot

4

adj

opp

Ѳ

5

4

3

Sol: Given that cosѲ = 3/5 = adj / hyp

thus using reference triangle adj = 3, hyp = 5,by Pythagorean principle opp = 4

THANK YOU