Find the distance between the points `(a cos 60^(@), 0)` and `(0, a sin 60^(@))`.

Answer

To find the distance between the points \((a \cos 60^\circ, 0)\) and \((0, a \sin 60^\circ)\), we will use the distance formula. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:

\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]

### Step 1: Identify the coordinates
The given points are:
- Point 1: \((x_1, y_1) = (a \cos 60^\circ, 0)\)
- Point 2: \((x_2, y_2) = (0, a \sin 60^\circ)\)

### Step 2: Substitute the coordinates into the distance formula
Using the coordinates in the distance formula, we have:

\[ d = \sqrt{(0 - a \cos 60^\circ)^2 + (a \sin 60^\circ - 0)^2} \]

### Step 3: Simplify the expression
This simplifies to:

\[ d = \sqrt{(-a \cos 60^\circ)^2 + (a \sin 60^\circ)^2} \]

### Step 4: Calculate the squares
Calculating the squares gives:

\[ d = \sqrt{(a^2 \cos^2 60^\circ) + (a^2 \sin^2 60^\circ)} \]

### Step 5: Factor out \(a^2\)
We can factor out \(a^2\):

\[ d = \sqrt{a^2 (\cos^2 60^\circ + \sin^2 60^\circ)} \]

### Step 6: Use the Pythagorean identity
Using the trigonometric identity \(\cos^2 \theta + \sin^2 \theta = 1\):

\[ d = \sqrt{a^2 \cdot 1} \]

### Step 7: Simplify the square root
This simplifies to:

\[ d = \sqrt{a^2} = a \]

### Final Answer
Thus, the distance between the points \((a \cos 60^\circ, 0)\) and \((0, a \sin 60^\circ)\) is:

\[ \boxed{a} \]