If the sides of a right angled triangle are in A.P then the sines of the acute angles are

Options

A. `3//5,4//5`
B. `sqrt(3),1//sqrt(3)`
C. `sqrt((sqrt(5)-1)/(2)),sqrt((sqrt(5)+1)/(2))`
D. `sqrt((sqrt(3)-1)/(2)),sqrt((sqrt(3)+1)/(2))`

Answer

A

Explanation

Let ABC be the right triangle, right angled at B. Let the sides of the triangle be a-d,a and a+d, with `agtdgt0`.
Clearly, AC=a+d is the largest side. So, a+d is the side opposite to the right angle.
By Pythagoras theorem, we have
`(a+d)^(2)=a^(2)+(a-d)^(2)rArr4ad=a^(2)rArra=4d`.
Now, `sinA=(BC)/(AC)=(a)/(a+d)=(4d)/(5d)=(4)/(5)`
`and,sinC=(AB)/(AC)=(a-d)/(a+d)=(4d-d)/(4d+d)=(3)/(5)`