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

How to find the general and principal values of cos$^{-1}$ x?

Let cos θ = x where, (- 1 ≤ x ≤ 1) then θ = cos$^{-1}$ x.

Here θ has infinitely many values.

Let 0 ≤ α ≤ $\frac{π}{2}$, where α is positive smallest numerical value and satisfies the equation cos θ = x then the angle α is called the principal value of cos$^{-1}$ x.

Again, if the principal value of cos$^{-1}$ x is α (0 ≤ α ≤ π) then its general value = 2nπ ± α

Therefore, cos$^{-1}$ x = 2nπ ± α, where, 0 ≤ α ≤ π and (- 1 ≤ x ≤ 1).

Examples to find the general and principal values of arc cos x:

1. Find the General and Principal Values of cos$^{-1}$ ½

Solution:

Let x = cos$^{-1}$ ½

⇒ cos x = ½

⇒ cos x = cos $\frac{π}{3}$

⇒ x = $\frac{π}{3}$

⇒ cos$^{-1}$ ½ = $\frac{π}{3}$

Therefore, principal value of cos$^{-1}$ ½ is $\frac{π}{3}$ and its general value = 2nπ ± $\frac{π}{3}$.

2. Find the General and Principal Values of cos$^{-1}$ (-½)

Solution:

Let x = cos$^{-1}$ (-½)

⇒ cos x = (-½)

⇒ cos x = - cos $\frac{π}{3}$

⇒ cos x = cos (π - $\frac{π}{3}$)

⇒ x = $\frac{2π}{3}$

⇒ cos$^{-1}$ (-½) = $\frac{2π}{3}$

Therefore, principal value of cos$^{-1}$ (-½) is $\frac{2π}{3}$ and its general value = 2nπ ± $\frac{2π}{3}$.

● 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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