![𝒔𝒊𝒏−𝟏 𝒙
▶ Domain : [-
1,1]
▶ Range : [-
π
,π
]
2 2
▶ Increasing
function
▶ Odd function
▶ sin−1( −𝑥) = -
sin−1 𝑥
▶ sin−1
𝑥=
𝑑 1
𝑑𝑥
1−𝑥2](https://image.slidesharecdn.com/itf-converted-241125002234-c765d797/85/ITF-converted-pptx-class-12-cbse-project-3-320.jpg)
![𝒄𝒐𝒔−𝟏 𝒙
▶ Domain : [-1,1]
▶ Range : [0, π]
▶ Decreasing function
▶ Neither odd nor even
▶ cos−1( −𝑥 )= π -
cos−1 𝑥
𝑑
𝑥
▶
𝑑
cos−1 𝑥
= -
1
1−𝑥2](https://image.slidesharecdn.com/itf-converted-241125002234-c765d797/85/ITF-converted-pptx-class-12-cbse-project-4-320.jpg)
![𝒔𝒆𝒄−𝟏 𝒙
▶ Domain : −∞, −1 𝑈 [1,
∞)
2
2
▶ Range : [0,π
) U
(𝜋
,π]
▶ Neither even nor odd
function
▶ sec−1(−𝑥) = π - sec−1 𝑥
𝑑
𝑥
▶
𝑑
sec−1 𝑥
=
1
𝑥
𝑥2−1](https://image.slidesharecdn.com/itf-converted-241125002234-c765d797/85/ITF-converted-pptx-class-12-cbse-project-7-320.jpg)
![𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙
▶ Domain : −∞. −1 U [1,
∞)
▶ Range : [
�
�
2
− , 0) U
(0,
2
𝜋
]
▶ Odd function
▶ csc−1 −𝑥=
−csc−1 𝑥
𝑑
𝑥
▶
𝑑
csc−1 𝑥
= -
1
𝑥
𝑥2−1](https://image.slidesharecdn.com/itf-converted-241125002234-c765d797/85/ITF-converted-pptx-class-12-cbse-project-8-320.jpg)
ITF-converted.pptx class 12 cbse project
- 2. INTRODUCTION ▶ Onto function : (aka surjective function) - Range and Co-Domain of a function are same, for eg: For a function to be inverse trigo function, it needs to be 1-1 and onto function (aka bijective function) ▶ One to one function: (aka injective function) -No two elements have same output, for eg:
- 3. 𝒔𝒊𝒏−𝟏 𝒙 ▶ Domain : [- 1,1] ▶ Range : [- π ,π ] 2 2 ▶ Increasing function ▶ Odd function ▶ sin−1( −𝑥) = - sin−1 𝑥 ▶ sin−1 𝑥= 𝑑 1 𝑑𝑥 1−𝑥2
- 4. 𝒄𝒐𝒔−𝟏 𝒙 ▶ Domain : [-1,1] ▶ Range : [0, π] ▶ Decreasing function ▶ Neither odd nor even ▶ cos−1( −𝑥 )= π - cos−1 𝑥 𝑑 𝑥 ▶ 𝑑 cos−1 𝑥 = - 1 1−𝑥2
- 5. 𝒕𝒂𝒏−𝟏 𝒙 ▶ Domain : All Real no.s π ▶ Range : (-π ,) 2 2 ▶ Increasing function ▶ Odd function ▶ tan−1 −𝑥 = −tan−1 𝑥 𝑑 𝑥 ▶ 𝑑 tan−1 𝑥 = 1 1+𝑥2
- 6. 𝒄𝒐𝒕−𝟏 𝒙 ▶ Domain : All Real No.s ▶ Range : (0, π) ▶ Decreasing function ▶ Neither odd nor even function ▶ cot−1(−𝑥)= π - cot−1 𝑥 𝑑 𝑥 ▶ 𝑑 cot−1(𝑥)= − 1 1+𝑥2
- 7. 𝒔𝒆𝒄−𝟏 𝒙 ▶ Domain : −∞, −1 𝑈 [1, ∞) 2 2 ▶ Range : [0,π ) U (𝜋 ,π] ▶ Neither even nor odd function ▶ sec−1(−𝑥) = π - sec−1 𝑥 𝑑 𝑥 ▶ 𝑑 sec−1 𝑥 = 1 𝑥 𝑥2−1
- 8. 𝒄𝒐𝒔𝒆𝒄−𝟏 𝒙 ▶ Domain : −∞. −1 U [1, ∞) ▶ Range : [ � � 2 − , 0) U (0, 2 𝜋 ] ▶ Odd function ▶ csc−1 −𝑥= −csc−1 𝑥 𝑑 𝑥 ▶ 𝑑 csc−1 𝑥 = - 1 𝑥 𝑥2−1
- 9. NOTE: ▶ None of the inverse trigo functions are periodic functions. ▶ 𝑓(f −1 𝑥 ) = x for every x belonging to its domain ▶ Sign of an inequality changes if we apply a decreasing function on both sides of an inequality. Sign of inequality remains same if it's an increasing function. ▶ 𝑦 = sin−1(sin 𝑥) = nπ + (−1)𝑛 𝑥 ▶ 𝑦 = cs𝑐−1 (csc 𝑥) = nπ + (−1)𝑛 𝑥
- 10. ▶ 𝑦 = cos−1 (cos 𝑥) = 2nπ ± 𝑥 ▶ 𝑦 = sec−1 (sec 𝑥) = 2nπ ± 𝑥
- 11. ▶ 𝑦 = tan−1(tan 𝑥) = 𝑛𝜋 + 𝑥 ▶ 𝑦 = cot−1(𝑐𝑜𝑡 𝑥) = 𝑛𝜋 + 𝑥
- 12. PROPERTIES: ▶ sin−1 𝑥 + cos−1 𝑥 = � � ▶ tan−1 𝑥 + co𝑡−1 𝑥 = 2 � � 2 ▶ s𝑒𝑐−1 𝑥 + cosec−1 𝑥 = � � 2 � � ▶ sin−1 1 = csc−1 𝑥 � � ▶ cos−1 1 = sec−1 𝑥 ▶ tan −1 1 𝑥 cot−1 𝑥 = ቊ −𝜋 + cot−1 𝑥 ; 𝑥 > 0 ; 𝑥 < 0 � � � � ▶ tan−1 𝑥 + tan−1 1 = ቐ2 − 𝜋 2 ; 𝑥 > 0 ; 𝑥 <
- 13. ▶ tan−1 𝑥 + tan−1 𝑦 = tan− 1 𝑥+ 𝑦 1−𝑥 𝑦 ; 𝑥 > 0, 𝑦 > 0, 𝑥𝑦 < 1 𝜋 + tan−1 𝑥+ 𝑦 1−𝑥 𝑦 ; 𝑥 > 0, 𝑦 > 0, 𝑥𝑦 > 1 ▶ tan−1 𝑥 − tan−1 𝑦 = tan−1 𝑥− 𝑦 1+𝑥 𝑦 ; 𝑥 > 0, 𝑦 > 0
- 14. THANKYOU