A vector (x/y)  has magnitude 5 units and direction tan-¹ (3/4). Find the values of x and y.

To find the values of \( x \) and \( y \) for a vector with magnitude \( 5 \) units and direction \( \theta = \tan^{-1} \left( \frac{3}{4} \right) \), we can use the following steps:

1. **Determine the angle \( \theta \):**
   \[
   \theta = \tan^{-1} \left( \frac{3}{4} \right)
   \]

2. **Find \( \cos \theta \) and \( \sin \theta \):**

   Since \(\theta = \tan^{-1} \left( \frac{3}{4} \right)\), we know:
   \[
   \tan \theta = \frac{3}{4}
   \]
   
   From trigonometric identities and the definition of tangent:
   \[
   \tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{3}{4}
   \]
   
   We can form a right triangle with opposite side \( 3 \), adjacent side \( 4 \), and hypotenuse \( 5 \) (from the Pythagorean theorem).

   Thus:
   \[
   \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5}
   \]
   \[
   \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5}
   \]

3. **Use the magnitude and direction to find \( x \) and \( y \):**
   The vector \( \

Learn how to calculate the components of a vector with magnitude 5 units and direction tan^-1(3/4). This problem involves trigonometric ratios and vector analysis, suitable for high school students.

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