(i) The angle of a triangle are in A.P and the greatest angle of the triangle is double the least angle , find the greatest angle in radians.(ii) The angle of a traingle are in A.P and the number of degree in the least to the number of radian in the greatest is as `60^(pi)` find the angles in degree and radians.
(iii) The angles of a traingle are in A.P and one of them is `80^(@)`. Find all the angle in sexagesimal system.
(iv)The angle of a traingle are in A.P and the greatest is `84^(@)`. Find all the three angle in radians.

Answer

Let's solve the questions step by step.

### Part (i)
**Given:** The angles of a triangle are in Arithmetic Progression (A.P.) and the greatest angle is double the least angle.

1. Let the least angle be \( x \).
2. Then, the greatest angle is \( 2x \).
3. The middle angle can be expressed as \( x + d \), where \( d \) is the common difference.
4. The angles of the triangle are \( x \), \( x + d \), and \( 2x \).
5. The sum of the angles in a triangle is \( 180^\circ \):
\[ x + (x + d) + 2x = 180^\circ \]
\[ 4x + d = 180^\circ \]

6. Since the angles are in A.P., the middle angle can also be expressed as:
\[ d = 2x - x = x \]
Thus, substituting \( d \) in the equation gives:
\[ 4x + x = 180^\circ \implies 5x = 180^\circ \implies x = \frac{180^\circ}{5} = 36^\circ \]

7. The greatest angle is:
\[ 2x = 2 \times 36^\circ = 72^\circ \]

8. To convert \( 72^\circ \) to radians:
\[ \text{Radians} = 72^\circ \times \frac{\pi}{180^\circ} = \frac{72\pi}{180} = \frac{2\pi}{5} \]

**Greatest angle in radians:** \( \frac{2\pi}{5} \)