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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