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

• General and Principal Values of sin$^{-1}$ x
• General and Principal Values of cos$^{-1}$ x
• General and Principal Values of tan$^{-1}$ x
• General and Principal Values of csc$^{-1}$ x
• General and Principal Values of sec$^{-1}$ x
• General and Principal Values of cot$^{-1}$ x
• Principal Values of Inverse Trigonometric Functions
• General Values of Inverse Trigonometric Functions
  
• arcsin(x) + arccos(x) = $\frac{π}{2}$
• arctan(x) + arccot(x) = $\frac{π}{2}$
• arctan(x) + arctan(y) = arctan($\frac{x + y}{1 - xy}$)
• arctan(x) - arctan(y) = arctan($\frac{x - y}{1 + xy}$)
• arctan(x) + arctan(y) + arctan(z)= arctan$\frac{x + y + z – xyz}{1 – xy – yz – zx}$
• arccot(x) + arccot(y) = arccot($\frac{xy - 1}{y + x}$)
• arccot(x) - arccot(y) = arccot($\frac{xy + 1}{y - x}$)
• arcsin(x) + arcsin(y) = arcsin(x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)
• arcsin (x) - arcsin(y) = arcsin (x $\sqrt{1 - y^{2}}$ - y$\sqrt{1 - x^{2}}$)
• arccos (x) + arccos(y) = arccos(xy - $\sqrt{1 - x^{2}}$$\sqrt{1 - y^{2}}$)
• arccos(x) - arccos(y) = arccos(xy + $\sqrt{1 - x^{2}}$$\sqrt{1 - y^{2}}$)
• 2 arcsin(x) = arcsin(2x$\sqrt{1 - x^{2}}$) 
• 2 arccos(x) = arccos(2x$^{2}$ - 1)
• 2 arctan(x) = arctan($\frac{2x}{1 - x^{2}}$) = arcsin($\frac{2x}{1 + x^{2}}$) = arccos($\frac{1 - x^{2}}{1 + x^{2}}$)
• 3 arcsin(x) = arcsin(3x - 4x$^{3}$)
• 3 arccos(x) = arccos(4x$^{3}$ - 3x)
• 3 arctan(x) = arctan($\frac{3x - x^{3}}{1 - 3 x^{2}}$)
• Inverse Trigonometric Function Formula
• Principal Values of Inverse Trigonometric Functions
• Problems on Inverse Trigonometric Function

11 and 12 Grade Math

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