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

Answer

A

Explanation

Let the sides of right angled triangle be
`(a-d),(a+d)(agtd)`.
By Pythagoras theorem,
`(a+d)^(2)=a^(2)+(a-d)^(2)`
`a^(2)+d^(2)+2ad=a^(2)+a^(2)+d^(2)-2ad`
`a^(2)=4ad`
`a=4d ' " [" since " ane 0]"……(i)"`
According ti the question, `sinA=(a)/((a+d)=(4d)/(5d)=(4)/(5)`
`sinC=(a-d)/(a+d)=(3d)/(5d)=(3)/(5)`.