A line makes the same angle with each of the x and z -axes. If the angle , which it makes with the y -axis is such that ^2 =3 ^2 , then ^2 equals

Step-by-step solution

Let the direction cosines of the line be l, m, n . Since the line makes the same angle with the x and z -axes, we have l = and n = . Let be the angle the line makes with the y -axis, so m = . The sum of the squares of the direction cosines is l^2 + m^2 + n^2 = 1 . Substituting the values, we get ^2 + ^2 + ^2 = 1 , which simplifies to 2 ^2 + ^2 = 1 . We know that ^2 = 1 - ^2 and ^2 = 1 - ^2 . Given the condition ^2 = 3 ^2 , we substitute the identities: 1 - ^2 = 3(1 - ^2 ) . From the relation 2 ^2 + ^2 = 1 , we have