## Trigonometry

#### Trigonometry

Trigonometry is composed of two words. The first word trigono , which means - the triangle and the second word metry means - measurement. That is, the word trigonometry literally means - the science of triangle measurement or related to the sides and angles of the triangle.
In modern times , it is defined as that branch of mathematics which measuring the sides of the triangle and establishing relationships between the sides and the angles.
Now a days , trigonometry is used in various fields such as surveying , astronomy , navigation , physics , engineering etc.

### Important Formulae and Results of Trigonometry:-

( Ⅰ )

1. 180° = π radian

 2. 1° = π = 0.01745 radian 180

 3. π = Circumference of a circle Diameter of the circle

 π = 22 = 3.141 ( Approx. ) 7

 4. Angle = Arc ⇒ θ = L Radius r

 5. Each interior angle of a regular polygon of n sides = ( n - 2 ) x 180° n

6. 1 right angle = 90°
1° = 60' ( 60 minutes ) and 1' = 60'' ( 60 seconds )

### Trigonometric Ratios :-

( Ⅱ ) 1. Sinθ × Cosecθ = 1

 ⇒ Sinθ = AB = 1 BC Cosecθ

 ⇒ Cosecθ = BC = 1 AB Sinθ

2. Secθ × Cosθ = 1

 ⇒ Cosθ = AC = 1 BC Secθ

 ⇒ Secθ = BC = 1 AC Cosθ

-1 ≤ Cosθ ≤ 1 , Secθ ≤ -1 या Secθ ≥ 1

3. tanθ × Cotθ = 1

 ⇒ Cotθ = AC = 1 AB tanθ

 ⇒ tanθ = AB = 1 AC Cotθ

-∞ < tanθ < ∞ , -∞ < Cotθ < ∞

Note :- We can use [ PBP / HHB ] for finding the values of trigonometric ratios. where , P = Perpendicular , B = Base and H = Hypotenuse

### Trigonometric Identities :-

1. Sin2θ + Cos2θ = 1
⇒ Sin2θ = 1 - Cos2θ
⇒ Cos2θ = 1 - Sin2θ

2. Sec2θ - tan2θ = 1
⇒ Sec2θ = 1 + tan2θ
⇒ tan2θ = Sec2θ - 1

3. Cosec2θ - Cot2θ = 1
⇒ Cosec2θ = 1 + Cot2θ
⇒ Cot2θ = Cosec2θ - 1

 4. tanθ = Sinθ ; Cosθ

 Cotθ = Cosθ Sinθ

### Pythagoras Theorem :-

According to Pythagoras theorem , In right angle ΔABC ,( fig. 1 . )
( Hypotenuse )2 = ( Base )2 + ( Perpendicular )2
From figure , Hypotenuse = BC , Base = AC and Perpendicular = AB
( BC )2 = ( AC )2 + ( AB )2
⇒ BC = √( AC )² + ( AB )²

### Trigonometric Ratios of Some specific angles

 ( ⅰ ) Sin15° = ( √3 - 1 ) 2√2

 Cos15° = ( √3 + 1 ) 2√2

tan15° = 2 - √3

 ( ⅱ ) Sin18° = ( √5 - 1 ) = Cos72° 4

 Cos18° = ( √10 + 2√5 ) = Sin72° 4

 ( ⅲ ) Cos36° = ( √5 + 1 ) = Sin54° 4

 Sin36° = ( √10 - 2√5 ) = Cos54° 4

 ( ⅳ ) tan7 1° = ( √3 - √2 )( √2 - 1 ) 2

 Cot7 1° = ( √3 + √2 )( √2 + 1 ) 2

### Trigonometric ratios of complementary and supplementary angles of angle θ :-

#### Complementary angle :-

If the sum of any two given angles are 90° , then these angles are called complementary angle .

#### ( ⅰ ) Trigonometric ratios of complementary angle :-

• Sin(-θ) = -Sinθ
• Cos(-θ) = Cosθ
• tan(-θ) = -tanθ
• Cot(-θ) = -Cotθ
• Sec(-θ) = Secθ
• Cosec(-θ) = -Cosecθ
• Sin(90° - θ ) = Cosθ
• Cos(90° - θ ) = Sinθ
• tan(90° - θ ) = Cotθ
• Cot(90° - θ ) = tanθ
• Sec(90° - θ ) = Cosecθ
• Cosec(90° - θ ) = Secθ
• Sin(90° + θ ) = Cosθ
• Cos(90° + θ ) = -Sinθ
• tan(90° + θ ) = -Cotθ
• Cot(90° + θ ) = -tanθ
• Sec(90° + θ ) = -Cosecθ
• Cosec(90° + θ ) = Secθ

#### Supplementary angle :-

If the sum of any two given angles are 180° , then these angles are called supplementary angle .

#### ( ⅱ ) Trigonometric ratios of supplementary angle :-

• Sin(180° - θ ) = Sinθ
• Cos(180° - θ ) = -Cosθ
• tan(180° - θ ) = -tanθ
• Cot(180° - θ ) = -Cotθ
• Sec(180° - θ ) = -Secθ
• Cosec(180° - θ ) = Cosecθ
• Sin(180° + θ ) = -Sinθ
• Cos(180° + θ ) = -Cosθ
• tan(180° + θ ) = tanθ
• Cot(180° + θ ) = Cotθ
• Sec(180° + θ ) = -Secθ
• Cosec(180° + θ ) = -Cosecθ
• Sin(270° - θ ) = -Cosθ
• Cos(270° - θ ) = -Sinθ
• tan(270° - θ ) = Cotθ
• Cot(270° - θ ) = tanθ
• Sec(270° - θ ) = -Cosecθ
• Cosec(270° - θ ) = -Secθ
• Sin(270° + θ ) = -Cosθ
• Cos(270° + θ ) = Sinθ
• tan(270° + θ ) = -Cotθ
• Cot(270° + θ ) = -tanθ
• Sec(270° + θ ) = Cosecθ
• Cosec(270° + θ ) = -Secθ
• Sin(360° - θ ) = -Sinθ
• Cos(360° - θ ) = Cosθ
• tan(360° - θ ) = -tanθ
• Cot(360° - θ ) = -Cotθ
• Sec(360° - θ ) = Secθ
• Cosec(360° - θ ) = -Cosecθ
• Sin(360° + θ ) = -Sinθ
• Cos(360° + θ ) = Cosθ
• tan(360° + θ ) = tanθ
• Cot(360° + θ ) = Cotθ
• Sec(360° + θ ) = Secθ
• Cosec(360° + θ ) = -Cosecθ

#### Trigonometric formulae for the Sum / Difference of two angles :-

( ⅰ ) Sin( A + B ) = SinA.CosB + CosA.SinB
( ⅱ ) Sin( A - B ) = SinA.CosB - CosA.SinB
( ⅲ ) Cos( A + B ) = CosA.CosB - SinA.SinB
( ⅳ ) Cos( A - B ) = CosA.CosB + SinA.SinB

 ( ⅴ ) tan( A + B ) = ( tanA + tanB ) ( 1 - tanA.tanB )

 ( ⅵ ) tan( A - B ) = ( tanA - tanB ) ( 1 + tanA.tanB )

 ( ⅶ ) Cot( A + B ) = ( CotA.CotB - 1 ) ( CotB + CotA )

 ( ⅷ ) Cot( A - B ) = ( CotA.CotB + 1 ) ( CotB - CotA )

 ( ⅸ ) tan( A + B + C ) = ( tanA + tanB + tanC - tanA.tanB.tanC ) [ 1 - ( tanA.tanB + tanB.tanC + tanC.tanA ) ]

( ⅹ ) Sin( A + B ).Sin( A - B ) = Sin2A - Sin2B = Cos2B - Cos2A
( ⅺ ) Cos( A + B ).Cos( A - B ) = Cos2A - Sin2B = Cos2B - Sin2A

#### Formulae of Sum / Difference of Sine and Cosine :-

 ( ⅰ ) sinC + sinD = 2sin (C + D) .cos (C - D) 2 2

 ( ⅱ ) sinC - sinD = 2cos (C + D) .sin (C - D) 2 2

 ( ⅲ ) cosC + cosD = 2cos (C + D) .cos (C - D) 2 2

 ( ⅳ ) cosC - cosD = 2sin (C + D) .sin (D - C) 2 2

( ⅴ ) 2sinA.cosB = sin( A + B ) + sin( A - B )

( ⅵ ) 2cosA.sinB = sin( A + B ) - sin( A - B )

( ⅶ ) 2cosA.cosB = cos( A + B ) + cos( A - B )

( ⅷ ) 2sinA.sinB = cos( A - B ) - cos( A + B )

#### Trigonometric ratios of multiple angles :-

 ( ⅰ ) sin2A = 2sinA.sinB = 2tanA ( 1 + tan2A )

 ( ⅱ ) cos2A = cos2A - sin2A = 2cos2A - 1 = 1 - 2sin2A = ( 1 - tan2A ) ( 1 + tan2A )

 ( ⅲ ) tan2A = 2tanA ( 1 + tan2A )

 ( ⅳ ) sin2A = ( 1 - cos2A ) 2

 cos2A = ( 1 + cos2A ) 2

( ⅴ ) sin3A = 3sinA - 4sin3A

( ⅵ ) cos3A = 4cos3A - 3cosA

 ( ⅶ ) cot3A = ( cot3A - 3cotA ) ( 3cot2A - 1 )

 ( ⅷ ) tan3A = ( 3tanA - tan3A ) ( 1 - 3tan2A )

 ( ⅸ ) tanA = √ ( 1 - cos2A ) = ( 1 - cos2A ) ( 1 + cos2A ) sin2A

#### Trigonometric ratios of sub-multiple angles :-

 ( ⅰ ) sinA = 2sin A cos A 2 2

 sinA = 2tan( A/2 ) [ 1 + tan2( A/2 ) ]

 ( ⅱ ) cosA = cos2 A - sin2 A = 2cos2 A - 1 = 1 - 2sin2 A 2 2 2 2

 cosA = [ 1 - tan2( A/2 ) ] [ 1 + tan2( A/2 ) ]

 ( ⅲ ) tanA = 2tan( A/2 ) [ 1 + tan2( A/2 ) ]

 ( ⅳ ) sin2 A = ( 1 - cosA ) 2 2

 cos2 A = ( 1 + cosA ) 2 2

 ( ⅴ ) 2sin A = ± √1 + sinA ± √1 - sinA 2

 ( ⅵ ) 2cos A = ± √1 + sinA ∓ √1 - sinA 2

 ( ⅶ ) tan A = √ ( 1 - cosA ) = ( 1 - cosA ) 2 ( 1 + cosA ) sinA

Example 1 . What is the value of 150° ( in radian ) ?

 Sol :- ∵ 1 radian = 180° π

 ∴ 1° = π radian 180

 ⇒ 150° = 150 x π radian 180

 ⇒ 150° = 5π radian 6

 Example 2 . If cos A = 4 , then find the value of sin A . 5

Sol :- We know that , sin2A + cos2A = 1
⇒ sin2A = 1 - cos2A

 ⇒ sin2A = 1 - 4 2 5

 ⇒ sin2A = 1 - 16 = ( 25 - 16 ) 25 25

 = 9 25

 ∴ sin A = √ 9 = 3 25 5

 Example 3 . If α + θ = 7π and tanθ = √3 , then what will be the value of tanα ? 12

Sol :- Given , tanθ = √3 = tan60°

 ⇒ θ = 60° = π 3

 ∵ α + θ = 7π 12

 ⇒ α + π = 7π 3 12

 ⇒ α = 7π - π 12 3

 ⇒ α = (7π - 4π) = 3π = π = 45° 12 12 4

⇒ α = 45°
∴ tanα = tan45° = 1

Example 4 . If , where 3θ and (θ - 2) are acute angles , then what is the value of θ ?
Sol :- Given that , sin3θ = cos(θ - 2)
⇒ sin3θ = sin[90° - (θ - 2)]
⇒ 3θ = [90° - (θ - 2)]
⇒ 4θ = 90° + 2 ⇒ 4θ = 92°

 ⇒ θ = 92° = 23° 4

Hence the value of θ will be 23° .

Example 5 .The value of tan1°.tan2°.tan3°………tan88°.tan89° is -
Sol :- tan1°.tan2°.tan3°………tan88°.tan89° =
tan1°.tan2°.tan3°………tan(90° - 2°).tan(90° - 1°)
= tan1°.tan2°.tan3°………cot2° - .cot1°
= tan1°.cot1°.tan2°.cot2°…….tan45°
= 1.1………tan45° = tan45° = 1 { ∴ tanA°.cotA° = 1 }
∴ tan1°.tan2°.tan3°………tan88°.tan89° = 1

Example 6 . Find the value of cos210° .
Sol :- We know that cos(180° + θ) = -cosθ
cos210° = cos(180° + 30°)

 -cos30° = -√3 2

Example 7 . Find the value of sin22° + sin24° + sin26° + …………+ sin288° + sin290°
Sol :- sin22° + sin24° + sin26° + …………+ sin288° + sin290° = sin22° + sin24° + sin26° + …………+ sin2(90° - 2°) + sin290° [ ∴ sin2(90° - 2°) = cos22° ]
= sin22° + sin24° + sin26° + …………+ cos22° + 1
= ( sin22° + cos22° ) + ( sin24° + cos24° ) + …………. + 1
= 1 + 1 + …….. upto 22 terms + 1 = 22 + 1
∴ sin22° + sin24° + sin26° + …………+ sin288° + sin290° = 23