# General and Principal Values of sin$$^{-1}$$ x

What are the general and principal Values of sin$$^{-1}$$ x?

What is sin$$^{-1}$$ ½?

We know that sin (30°) = ½.

⇒ sin$$^{-1}$$ (1/2) = 30° or $$\frac{π}{6}$$.

Again, sin θ = sin (π - $$\frac{π}{6}$$)

⇒ sin θ = sin ($$\frac{5π}{6}$$)

⇒ θ = $$\frac{5π}{6}$$or 150°

Again, sin θ = 1/2

⇒ sin θ = sin $$\frac{π}{6}$$

⇒ sin θ = sin (2π + $$\frac{π}{6}$$)

⇒ sin θ = sin ($$\frac{13π}{6}$$)

⇒ θ = $$\frac{13π}{6}$$ or 390°

Therefore, sin (30°) = sin (150°) = sin (390°) and so on, and, sin (30°) = sin (150°) = sin (390°) = ½.

In other ward we can say that,

sin (30° + 360° n) = sin (150° + 360° n) = ½, where, where n = 0, ± 1, ± 2, ± 3, …….

And in general, if sin θ = ½ = sin $$\frac{π}{6}$$ then θ = nπ + (- 1)$$^{n}$$ $$\frac{π}{6}$$, where n = 0 or any integer.

Therefore, if sin θ = 1/2 then θ = sin$$^{-1}$$ ½ = $$\frac{π}{6}$$ or $$\frac{5π}{6}$$ or $$\frac{13π}{6}$$

Therefore in general, sin$$^{-1}$$  (½) = θ = nπ + (-1) $$^{n}$$ $$\frac{π}{6}$$ and the angle nπ + (- 1)$$^{n}$$ $$\frac{π}{6}$$ is called the general value of sin$$^{-1}$$ ½.

The positive or negative least numerical value of the angle is called the principal value

In this case the $$\frac{π}{6}$$ is the least positive angle. Therefore, the principal value of sin$$^{-1}$$ ½ is $$\frac{π}{6}$$.

Let sin θ = x and - 1 ≤ x ≤ 1

x ⇒ sin {nπ + (- 1)$$^{n}$$ θ}, where n = 0, ± 1, ± 2, ± 3, …….

Therefore, sin$$^{-1}$$ x = nπ + (- 1)$$^{n}$$ θ, where n = 0, ± 1, ± 2, ± 3, …….

For the above equation we can say that sin$$^{-1}$$ x may have infinitely many values.

Let – $$\frac{π}{2}$$ ≤ α ≤ $$\frac{π}{2}$$, where α is positive or negative smallest numerical value and satisfies the equation sin θ = x then the angle α is called the principal value of sin$$^{-1}$$ x.

Therefore, the general value of sin$$^{-1}$$ x is nπ + (- 1)$$^{n}$$ θ, where n = 0, ± 1, ± 2, ± 3, …….

The principal value of sin$$^{-1}$$ x is α, where - $$\frac{π}{2}$$ ≤ α ≤ $$\frac{π}{2}$$ and α satisfies the equation sin θ = x.

For example, principal value of sin$$^{-1}$$ (-$$\frac{√3}{2}$$) is -$$\frac{π}{3}$$and its general value is nπ + (- 1)$$^{n}$$ ∙ (-$$\frac{π}{3}$$) = nπ - (- 1)$$^{n}$$ ∙ $$\frac{π}{3}$$.

Similarly, principal value of sin$$^{-1}$$ ($$\frac{√3}{2}$$) is ($$\frac{π}{3}$$) and its general value is nπ + (- 1)$$^{n}$$ ($$\frac{π}{3}$$) = nπ - (- 1)$$^{n}$$ ∙ $$\frac{π}{6}$$.

Inverse Trigonometric Functions