Question

The value of sin⁻¹(sin 5π/3) is......,Select Proper option from the given options.
(a) - π/3
(b) 5π/3
(c) π/3
(d) 2π/3

Answer 1

we have to find the value of sin^-1(sin5π/3).

we know , sin^-1(sinx) = x , for -π/2 ≤ x ≤ π/2

sin^-1(sin5π/3) = sin^-1[sin(2π - π/3)]

we know, sin(2π - A) = - sinA

so, sin^-1[sin(2π - π/3)] = sin^-1[-sin(π/3)]

we also know, sin^-1(-x) = -sin^-1x

so, sin^-1[-sin(π/3)] = -sin^-1(sinπ/3)

= -π/3. [as we know, sin^-1(sinx) = x , for -π/2 ≤ x ≤ π/2 ]

hence, option (a) is correct.

Answer 2

Dear Student,

Answer: Option a ( -π/3)

Solution:

As we know that principal value branch of sin⁻¹x is [-π/2,π/2]


so, if value of x lies in the given interval,we can write
${sin}^{ - 1} ( \sin(x)) = x \\ \\ if \: \: x \: \: lies \: in \: principal \: value \: branch$
here we can by analysing that 5π/3 does not lies between -π/2 to π/2

So,convert it into

${ \sin }^{ - 1} (sin \: \frac{5\pi}{3}) \\ = { \sin }^{ - 1} (sin \: 2\pi - \frac{\pi}{3}) \\ \\ since \: \sin(2\pi - x) = - \sin(x) \\ \\ = { \sin }^{ - 1} (sin \:( - \frac{\pi}{3}) )\\ since \: \: {sin}^{ - 1} ( - x) = - {sin}^{ - 1} x \\ \\ = - \frac{\pi}{3}$
Hopefully this helps you.