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

Some important values of Trigonometric Ratios :-

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 :-

Formulae of Sum / Difference of Sine and Cosine :-

Trigonometric ratios of multiple angles :-

Trigonometric ratios of sub-multiple angles :-

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