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Introduction
If sin x = 1/2, we can write one value of x = π/6.
If sin x = 1/3, i.e. x is not a well known angle, then we can write x = sin-1 1/3.
Similarly,
cos x = t ⇒ x = cos-1 t.
tan x = t ⇒ x = tan-1 t.
Rules for defined values of sin-1 x, cos-1 x
- y = sin-1 x:
Domain: x Є [-1, 1]
Range (principal value branch of sin-1 x)
y Є [-π/2, π/2]
- y = cos-1 x:
Domain: x Є [-1, 1]
Range (principal value branch of cos-1 x)
y Є [0, π]
- y = tan-1 x:
Domain: x Є R
Range (principal value branch of tan-1 x)
y Є (-π/2, π/2)
- y = cosec-1 x:
Domain: x Є (-∞, -1] ∪ [1, ∞)
Range (principal value branch of cosec-1 x)
y Є [-π/2, 0) ∪ (0, π/2)
- y = sec-1 x:
Domain: x Є (-∞, -1] ∪ [l, ∞)
Range (principal value branch of sec-1 x)
y Є [0, π/2) ∪ (π/2, π]
- y = cot-1 x:
Domain: X Є R
Range (principal value branch of cot-1 x)
y Є (0, π)
Note the similarity in principal value branch of sin-1 x, cosec-1 x, tan-1 x.
Interval for allowed values of y is known as principal value branch of that inverse function.
Important Results
Important Results (I):
- sin (sin-1x) = x, cos (cos-1x) = x, ......
- sin-1 sin θ = θ, cos-1 cos θ = θ
if θ allows the restrictions on y in the definition of corresponding inverse function.
e.g. sin-1 sin2π/3 2π/3 because 2π/3 does not lie in the principal value branch of sin-1 x.
Hence sin-1 sin2π/3 = sin-1 sin(π - π/3)sin-1 sinπ/3 = π/3. - sin-1(-x) = -sin-1x cos-1(-x) = π - cos-1x
cosec-1(-x) = -cosec-1x sec-1(-x) = π - sec-1x
tan-1(-x) = -tan-1x cot-1(-x) = π - cot-1x
Important Results (II):
- If x > 0, y > 0 then
- lf x > 0, y > 0 then
- If x > 0
Important Results (III):
Results (IV):
sin.-1 x + sin.-1 y= sin.-1 , when x ≥ 0, y ≥ 0, x2 + y2 ≤ 1
sin.-1 x + sin.-1 y= π - sin.-1 , when x ≥ 0, y ≥ 0, x2 + y2 > 1
Results (V):
sin.-1 x - sin.-1 y= sin.-1 , 0 ≤ y ≤ x
Results (VI):
cos.-1 x + cos.-1 y = cos.-1 , where x ≥ 0, y ≥ 0
cos.-1 x - cos.-1 y = cos.-1 , where 0 ≤ x, ≤ y
Thanks
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