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arcsin(x) + arccos(x) = π2

We will learn how to prove the property of the inverse trigonometric function arcsin(x) + arccos(x) = π2.

Proof: Let, sin1 x = θ

Therefore, x = sin θ

x = cos (π2 - θ), [Since, cos (π2 - θ) = sin θ]

⇒ cos1 x = π2 - θ

⇒ cos1 x= π2 - sin1 x, [Since, θ = sin1 x]

⇒ sin1 x + cos1 x = π2

Therefore, sin1 x + cos1 x = π2.             Proved.

Solved examples on property of inverse circular function sin1 x + cos1 x = π2.

1. Prove that sin1 45 + sin1 513 + sin1 1665 = π2

Solution:

sin1 45 + sin1 513 + sin1 1665

= (sin1 45 + sin1 513) + sin1 1665

= sin1(451(513)2)+5131(45)2) + sin1 1665

= sin1 (45 × 1213 + 513 × 35) + sin1 1665

= sin1 6365 + sin1 1665

= cos11(6365)2) + sin1 1665

= cos1 1665 + sin1 1665

= π/2, since sin1x+cos1x=π2

Therefore, sin1 45 + sin1 513 + sin1 1665 = π2.                  Proved.

 

2. Solve the trigonometric equation: sin1 5x + sin1 12x = π2

Solution:

sin1 5x + sin1 12x = π2

⇒ sin1 12x = π2 - sin1 5x

⇒ sin1 12x = cos1 5x, [Since, we know that, sin1 5x + cos1 5x = π2]

⇒ sin1 12x = sin1 x225x

12x = x225x

x225 = 12, [Since, x ≠ 0]

⇒ x2 - 25 = 144       

⇒ x2 = 144 + 25   

⇒ x2 = 169        

⇒ x = ± 13

The solution x = - 13 does not satisfy the given equation. 

Therefore the required solution is x = 13.

 Inverse Trigonometric Functions








11 and 12 Grade Math

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