Trigonometry
The arc length S of a circle is . Where is the radius and is the angle subtended at the centre and is in radian
The area of a sector of a circle is . Where is the radius and is the angle subtended at the centre and is in radian
The angle b/n hour hand and minute hand when the time is A hour B minutes is given by | | Basic definition Basic identities:- √ √ √ √ √ √ ASTC
rule:- The trigonometric ratios of , , & in terms of can easily remembered from the following rules
When the angle is or , the t-ratio changes to co-ratio and co-ratio changes to t-ratio
When the angle is or the trigonometric ratio remains same. In each case the sign must be determined by ASTC rule.
Trigonometric ratios of compound
angles:- ( ) ( ) ( ) ( ) ( ) A B c 𝜃 𝐴𝑑𝑗 𝑂𝑝𝑝 𝐻𝑦𝑝 I QUAD
All trigonometric functions are +ve
II QUAD
Sine & Cosec are +ve
III QUAD
tan & cot are +ve
IV QUAD Cos & Sec are +ve
( ) ( ) ( )
Trigonometric ratios of multiple angle
:-
Trigonometric ratios of multiple angle
:-
Trigonometric ratios of half
angles:- ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Some Important Formulae
√ √ √ √ √ √ Transformation formulae:- ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
Trigonometric ratios of some standard 0 √ √ √ √ √ √ 1 √ √ 0 0 1 √ √ √ √ √ √ 0 √ √ 0 1 0 √ √ √ √ √ √ 0 0
Inverse Trigonometric Function
Inverse function: If is a function from A to B, i.e. , then inverse function exists if and only if function is bijective (i.e. both one-one and onto)
Inverse of Sine function: Let then we define , where and . The value of lies in [ ] is called its principal value.
Example: 1) ( ) , 2) ( ) ,
3) (√ )
4) doesnot exist ( because 2>1)
Note: 1) is not possible because
2) ( ) can also be written as ( )
Inverse of Cosine function: Let then we define , where and . The value of lies in is called its principal value.
Example: 1) ( ) , 2) (√ ) ,
3) (
√ )
Inverse of Tangent function: Let then we define , where and . The value of lies in ( ) is called its principal value.
Example: 1) ( ) , 2) (√ ) , 3) ( ) y x (1,0) (–1,0) 2 – 2 y –1 (1,0) x 2 y /2 x –/2
Inverse of Cotangent function: Let then we define , where and . The value of
lies in ( ) is called its principal value.
Example: 1) ( ) , 2) (√ ) ,
3) ( )
Inverse of Secant function: Let then we define , where and . The value of lies in { } is called its principal value.
Example: 1) , 2) (√ ) ,
3) ( )
Inverse of Cosecant function: Let then we define , where and . The
value of lies in [ ] { } is called its principal value.
Example: 1) ( ) , 2) ( ) ,
3) (√ )
Note: Domain and range of the inverse trigonometric functions as follows
Functions Domain Range (Principal Branch Value)
( ) ( ) Problems:
1) Find the domain of the following
1. 2. ( ) 3. 4. ( ) 2) Find the principal values of the following
1. (√ ) 2. 3. √ 4. √ 5. √ y y = (0, /2) x y (0,) (0,/2) (–1,0) O (–1,0) x y (–1,0) O (1,0) x y = /2 y = –/2
Properties of inverse trigonometric functions: I. 1) ( ) where Proof: Let ( ) ( ) from (1) 2) ( ) where Proof: Let ( ) ( ) from (1) 3) ( ) where Proof: Let ( ) ( ) from (1)
Similarly we can prove that
4) ( ) where 5) ( ) where 6) ( ) where II 1) ( ) where Proof: Let ( ) ( ) from (1) 2) ( ) where Proof: Let ( ) ( ) from (1) 3) ( ) where Proof: Let ( ) ( ) from (1) Similarly we can prove that
4) ( ) where 5) ( ) where . 6) ( ) where Note : 1) – ( ) 2) – ( ) 3) – ( ) 4) – ( ) 5) – ( ) 6) – ( ) III 1) ( ) , where Proof: let ( ) ( ) ( ) ( ) from (1) 2) ( ) , where Proof: let ( ) ( ) ( ) ( ) from (1)
3) ( ) , where Proof: let ( ) ( ) ( ) ( ) from (1) 4) ( ) , where Proof: let ( ) ( ) ( ) ( ) from (1) 5) ( ) , where Proof: let ( ) ( ) ( ) ( ) from (1) 6) ( ) where
Proof: (home work)
IV 1) ( ) , where Proof: Let ( ) ( ) ( ) from (1) 2) ( ) , where Proof: Let ( ) ( ) ( ) from (1) 3) ( ) where Proof: Let ( ) ( ) ( ) from (1) Note: ( ) Proof: Let ..(1) ( ) ( ) ( ) ( ( )) ( )
Note: 1) ( ) 2) ( ) 3) ( ) 4) ( ) 5) ( ) 6) ( ) V 1) , where Proof : Let ( ) ( ) from (1) 2) , where Proof : Let ( ) ( ) from (1) 3) , where Proof : Let ( ) ( ) from (1) Problems:
1) Find the principal values of the following:
1) ( ) 2) ( √ ) 3) ( ) 4) ( √ ) 5) ( ) 6) (
√ ) 7) (
√ ) 2) Find the value of the following:
1) ( ) ( ) 2) (√ ) ( √ ) 3) (√ ) ( ) 4) ( ) ( ) ( ) 5) ( ) 6) ( ) 7) ( ) 8) ( ) 9) [ ( )] 10) ( ( )) 11) [ ] 12) [ ]
3) Prove the following:
1) ( ( )) √ 2) ( ) ( ) 3) 4) √ 5) (√ √ √ √ ) 6) ( )
Note:
Functions Substitution We Get
( ) ( ) √ √ √ √ √ √ √ √ √
4) Prove the following:
1) ( √ ) 2) ( √ ) 3) ( ) 4) ( ) 5) ( ) √ 6) √ 7) [ ( ) ( )] 8) [ ] 9) ( ) 5) Write the following functions in the simplest form:
1) √ , | | 2) √ | | 3) ( √ ) | | 4) ( ) √ √ 5) (√ ) 6) ( )
6) For what value of the following functions Satisfies 1) ( ) 2) ( ) 3) ( ) 4) ( √ ) 5) ( √ )
Some Standard Formulae:
1) ( ) Proof: Let Let Now, ( ) –( )( ) ( ) where ( ) ( ) ( ) Similarly we can prove:
2) ( ) 3. ( ) 4. ( ) | | Problems:
1) Prove the following: 1) 2) 3) 4) 5) [ ] ( ) 6) ( ) ( ) 7) 8) ( ) ( ) ( )
2) Solve the following equations:
1) ( ) 2) [ ] 3) 4) ( ) 5) [ ( ) ] 6) ( ) ( ) 𝛼 𝜋 𝑎𝑛𝑑 𝛽 𝜋 (𝛼 𝛽) 𝜋 𝜋 (𝛼 𝛽) 𝜋 (𝛼 𝛽) 𝑣𝑒 (𝛼 𝛽) 𝜋 Now, 𝑥𝑦 𝑥𝑦 𝑥𝑦 Now, 𝑥𝑦𝑥 𝑦 𝑣𝑒 𝑣𝑒 𝑣𝑒 ,
Home work
1) Find the values of the following: 1) ( √ √ ) ( √ √ ) 2) (√ ) ( √ ) 3) ( ) 4) ( ) ( ) 5) [ ] 6) ( ) 7) ( √ ) 8) 2) Prove the following:
1) ( ) 2) 3) (√ √ √ √ ) 4) ( ) 5) 6) √ 7) (√ √ √ √ ) 8) 9) 10)
Important Questions
One Mark Questions1) Write the domain of ( ) . 2) Write the domain of ( ) . 3) Write the domain of ( ) . 4) Write the domain of ( ) . 5) Write the domain of ( ) . 6) Write the domain of ( ) .
7) Write the range of . ( Write the principal value branch of ( ) ) 8) Write the range of .
9) Write the range of . 10) Write the range of . 11) Write the range of . 12) Write the range of .
13) If , then find the values of .
14) Write a range of ( ) other than [ ]. 15) Write a range of ( ) other than . 16) Find the principal value of ( √ ).
17) Find the principal value of ( ). 18) If ( √ ), then find the value of x. 19) Find the principal value of ( √ ). 20) Find the principal value of ( ).
21) Find the principal value of ( √ ). 22) Find the principal value of ( √ ). 23) Find the principal value of ( ). 24) Find the principal value of ( ).
25) Write the set of values of x for which ( ) . 26) Write the set of values of x for which ( ) ( ). 27) Write the set of values of x for which ( ). 28) Write the set of values of x for which ( ). 29) Write the set of values of x for which ( ) . 30) Write the set of values of x for which ( √ ) . 31) Write the set of values of x for which ( √ ) . 32) Find the value of ( ), | | .
33) Find the value of ( ). 34) Find the value of ( ).
35) Find the value of ( ). 36) Find the value of ( ).
Two Or Three Marks Questions: 1) Find the value of the following
1) [ ( )] 2) ( ) 3) ( )
4) ( ) 5) ( ) 6) √ ( ) 7) ( ( ))
2) Prove the following
1) 2) 3) ( √ ) √ √ 4) ( ) 5) ( ) [ ] 6) ( ) ( ) 7) ( ) [ ] 8) (√ √ √ √ ) 9) ( ) 10) ( ) 11) ( ) 12) ( ) 3) Write the following in simplest form
1) ( ) 2) [ ] 3) (√ ) 4) ( √ )
5) (√ ) 6) ( ) 7) [ ( ) ( )] 4) Prove that ( )
5) Prove the following 1) ( ) ( ) ( ) 2) ( ) [ ] | | √ 3) ( ) ( ) ( ) 4) 6) Find the value of of the following
1) ( ( ) ) 2) ( ) 3) ( ) ( ) 4) ( ) 5)
