Online Properties of a Inverse Trigonometric Function Assignment Help & Homework Help
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PROPERTIES OF INVERSE TRIGONOMETRIC FUNCTIONS::
Following are the properties of inverse trigonometric functions:
Proof of Property Number 1):
- sin(sin-1x) = x
Proof:
Let (sin-1x) = y
Hence sin(sin-1x) = sin y ⇒ x = sin y
Hence sin(sin-1x) = x
Similarly cos(cos-1x) = x, tan(tan-1x) = x, sec(sec-1x) = x, cosec(cosec-1x) = x and cot(cot -1x) = x can also be proved.
- sin-1(sin x) = x
Proof:
Let sin x = y
Hence sin-1(sin x) = sin-1y = 0
But sin x = y ⇒ sin-1y = x
Hence sin-1(sin x) = sin-1y = x
Similarly other parts of b) can also be proved.
Proof of Property Number 2):
sin-1
=
cosec-1x
Proof:
Let cosec-1x = y
Hence x = cosec y ⇒
= sin y ⇒ sin-1 =y
Hence, sin-1=
cosec-1x
Similarly other parts can also be proved.
Proof of Property Number 3):
sin-1(-x) = -sin-1x
Proof:
Let sin-1(-x) = y = 0
Hence -x = sin y
Or x = -sin y = sin(-y)
Hence x = sin(-y) ⇒ sin-1x = -y or y = -sin-1x
Similarly tan and cosec can also be proved.
cos-1(-x) = π - cos-1x
Proof:
Let cos-1(-x) = y
Hence -x = cos y or x = -cos y = cos(π - y)
Hence cos-1x = π - y ⇒ y = π - cos-1x
Hence, cos-1(-x) = π - cos-1x
Similarly sec and cot can also be proved.
Proof of Property Number 4):
sin-1x + cos-1x =
Proof:
Let sin-1x = y
Hence x = sin y = cos(- y)
Hence cos-1x =
- y or cos-1x-
sin-1x
Hence, sin-1x + cos-1x =
Similarly the other 2 parts can also be proved.
Proof of Property Number 5):
sin-1x ± sin-1y = sin-1[x√(1 - y2) ± y√(1 - x2)]
Proof:
Let sin-1x =
and sin-1y = β
Hence x = sin and y = sinβ
Then sin(± β) =
sincosβ ± cos
sinβ
= sin√(1 - sin2β) ±
sinβ√(1 - sin2)
= x√(1 - y2) ± y√(1 - x2)
Hence,± β
= sin-1[x√(1 - y2) ± y√(1 - x2)]
Thus, sin-1x ± sin-1y = sin-1[x√(1 - y2) ± y√(1 - x2)]
Similarly, the other two parts can also be proved.
Proof of Property Number 6):
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