Question

in a right triangle of abc , sin^A + sin^B + sin ^C is ------------------. mates
, please find it soon . I have the exam​

Answer 1

Answer:

Let the triangle ABC has the angle B as 90° .

When A and C are acute angles ( less than 90° ) we know by trigonometric relations that :

sin²A + cos²A = 1

sin²C + cos²C = 1

[ Proof : Take random value of A as 1/√2 when A = 45 .

( 1/√2 )² + ( 1/√2 )²

⇒ 1/2 + 1/2

⇒ 1 ]

Now in Δ ABC , we have :

∠A + ∠B + ∠C = 180

This is according to angle sum property which states that :-

Angle sum property of a triangle states that the sum of all angles of a triangle is 180 degrees .

We know that ∠B = 90° .

Hence :

∠A + ∠C + 90° = 180°

⇒ ∠A + ∠C = 90°

⇒ ∠A = 90° - ∠C

Now put the values we have got earlier :

sin²A + sin²B + sin²C

⇒ sin²A + sin²( 90 - C ) + sin² ( 90 )

By trigonometry we know sin 90 = 1 , also we know that sin² ( 90 - C ) = cos²A

We also know that sin²A + cos²A = 1 as proved earlier :

⇒ sin²A + cos²A + 1²

⇒ 1 + 1

⇒ 2

Hence the value will be 2 .

OPTION A is correct .