Question

The distance between the points (8 sin 60°, 0) and (0, 8 cos 60°) is

• A. 8
• B. 25
• C. 64
• D. \(\frac{1}{8}\)

Hint:

Before applying the distance formula, always simplify the coordinates as much as possible. This makes the calculation steps cleaner and reduces the chance of errors.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem requires finding the distance between two points in a Cartesian coordinate system. First, we need to evaluate the coordinates using trigonometric values, and then apply the distance formula.

Step 2: Key Formula or Approach:
The distance formula between two points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \] We also need the standard values: \(\sin 60^\circ = \frac{\sqrt{3}}{2}\) and \(\cos 60^\circ = \frac{1}{2}\).

Step 3: Detailed Explanation:
First, let's find the coordinates of the two points.
Point 1: \((x_1, y_1) = (8 \sin 60^\circ, 0) = (8 \times \frac{\sqrt{3}}{2}, 0) = (4\sqrt{3}, 0)\).
Point 2: \((x_2, y_2) = (0, 8 \cos 60^\circ) = (0, 8 \times \frac{1}{2}) = (0, 4)\).
Now, apply the distance formula:
\[ d = \sqrt{(0 - 4\sqrt{3})^2 + (4 - 0)^2} \] \[ d = \sqrt{(-4\sqrt{3})^2 + (4)^2} \] \[ d = \sqrt{(16 \times 3) + 16} \] \[ d = \sqrt{48 + 16} \] \[ d = \sqrt{64} \] \[ d = 8 \]

Step 4: Final Answer:
The distance between the two points is 8.