If the lengths of the sides of a right-angled triangle A B C , right angled at C , are in arithmetic progression, then the value of 5 sin ⁡ A + sin ⁡ B is

Step-by-step solution

a + d 2 = a - d 2 + a 2 (Pythagoras theorem) a 2 + d 2 + 2 a d = a 2 + d 2 - 2 a d + a 2 ⇒ 4 a d = a 2 ⇒ a = 4 d sin ⁡ A = a - d a + d = 4 d - d 4 d + d = 3 5 sin ⁡ B = a a + d = 4 d 5 d = 4 5 ⇒ 5 sin ⁡ A + 5 sin ⁡ B = 3 + 4 = 7 ⇒ 5 sin ⁡ A + sin ⁡ B = 7