arcsin (x) + arcsin(y) = arcsin (x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)

We will learn how to prove the property of the inverse trigonometric function arcsin (x) + arcsin(y) = arcsin (x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)

Proof:  

Let, sin$^{-1}$ x = α and sin$^{-1}$ y = β

From sin$^{-1}$ x = α we get,

x = sin α

and from sin$^{-1}$ y = β we get,

y = sin β

Now, sin (α + β) = sin α cos β + cos α sin β

⇒ sin (α + β) = sin α $\sqrt{1 - sin^{2} β}$ + $\sqrt{1 - sin^{2} α}$ sin β

⇒ sin (α + β) = x ∙ $\sqrt{1 - y^{2}}$ + $\sqrt{1 - x^{2}}$ ∙ y

Therefore, α + β = sin$^{-1}$ (x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$) 

or, sin$^{-1}$ x + sin$^{-1}$ y = sin$^{-1}$ (x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$).       Proved.

Note: If x > 0, y > 0 and x$^{2}$ + y$^{2}$ > 1, then the sin$^{-1}$ x + sin$^{-1}$ y may be an angle more than π/2 while sin$^{-1}$ (x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$), is an angle between – π/2 and π/2.

Therefore, sin$^{-1}$ x + sin$^{-1}$ y = π - sin$^{-1}$ (x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)

1. Prove that sin$^{-1}$ $\frac{3}{5}$ + sin$^{-1}$ $\frac{8}{17}$ = sin$^{-1}$ $\frac{77}{85}$

Solution:

L. H. S. = sin$^{-1}$ $\frac{3}{5}$ + sin$^{-1}$ $\frac{8}{17}$

Now, we will apply the formula sin$^{-1}$ x + sin$^{-1}$ y = sin$^{-1}$ (x $\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)

= sin$^{-1}$ ($\frac{3}{5}$ $\sqrt{1 - (\frac{8}{17})^{2}}$ + $\frac{8}{17}$$\sqrt{1 - (\frac{3}{5})^{2}}$)

= sin$^{-1}$ ($\frac{3}{5}$ ×  $\frac{15}{17}$ + $\frac{8}{17}$ ×  $\frac{4}{5}$)

= sin$^{-1}$  $\frac{77}{85}$ = R. H. S.                  Proved.

2. Show that, sin$^{-1}$ $\frac{4}{5}$ + sin$^{-1}$ $\frac{5}{13}$ + sin$^{-1}$ $\frac{16}{65}$ = $\frac{π}{2}$.  

Solution:     

L. H. S. = (sin$^{-1}$$\frac{4}{5}$ + sin$^{-1}$$\frac{5}{13}$) + sin$^{-1}$$\frac{16}{65}$

Now, we will apply the formula sin$^{-1}$ x + sin$^{-1}$ y = sin$^{-1}$ (x$\sqrt{1 - y^{2}}$ + y$\sqrt{1 - x^{2}}$)

= sin$^{-1}$ ($\frac{4}{5}$ $\sqrt{1 - (\frac{5}{13})^{2}}$ + $\frac{5}{13}$$\sqrt{1 - (\frac{4}{5})^{2}}$ + sin$^{-1}$$\frac{16}{65}$

= sin$^{-1}$ ($\frac{4}{5}$ ×  $\frac{12}{13}$ + $\frac{5}{13}$ ×  $\frac{3}{5}$) + sin$^{-1}$$\frac{16}{65}$

= sin$^{-1}$ $\frac{63}{65}$ + sin$^{-1}$$\frac{16}{65}$

= sin$^{-1}$ $\frac{63}{65}$ + cos$^{-1}$$\frac{63}{65}$, [Since, sin$^{-1}$ $\frac{16}{65}$ = cos$^{-1}$ $\frac{63}{65}$]

= $\frac{π}{2}$, [Since, sin$^{-1}$ x + cos$^{-1}$ x = $\frac{π}{2}$] = R. H. S.            Proved.

Note: sin$^{-1}$ = arcsin (x)

● 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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