In `DeltaABC, angleB=90^@), AB=y" units", BC=sqrt(3) " units"`, AC = 2 units and angle `A=x^(@)`, find :
use `cosx^(@)` to find the value of y.

Answer

To solve the problem step by step, we will use the properties of right triangles and trigonometric ratios.

### Step-by-Step Solution:

1. **Identify the Triangle and Given Values:**
- We have a right triangle \( \Delta ABC \) where \( \angle B = 90^\circ \).
- The sides are given as follows:
- \( AB = y \) units (opposite to angle A)
- \( BC = \sqrt{3} \) units (adjacent to angle A)
- \( AC = 2 \) units (hypotenuse)

2. **Use the Sine Function:**
- We know that:
\[ \sin A = \frac{\text{Opposite}}{\text{Hypotenuse}} = \frac{AB}{AC} \]
- For angle \( A \) (which is \( x \)):
\[ \sin x = \frac{AB}{AC} = \frac{y}{2} \]

3. **Calculate \( \sin x \) Using the Given Side BC:**
- We can also use the Pythagorean theorem to find \( y \):
\[ AC^2 = AB^2 + BC^2 \]
- Substituting the values:
\[ 2^2 = y^2 + (\sqrt{3})^2 \]
\[ 4 = y^2 + 3 \]
- Rearranging gives:
\[ y^2 = 4 - 3 = 1 \]
- Taking the square root:
\[ y = 1 \text{ (since length cannot be negative)} \]

4. **Using Cosine to Verify:**
- We can also find \( y \) using \( \cos x \):
\[ \cos x = \frac{\text{Adjacent}}{\text{Hypotenuse}} = \frac{BC}{AC} = \frac{\sqrt{3}}{2} \]
- We know that \( \cos 60^\circ = \frac{1}{2} \), so:
\[ \cos x = \frac{1}{2} \]
- From the cosine definition for angle \( A \):
\[ \cos x = \frac{y}{2} \]
- Setting these equal gives:
\[ \frac{y}{2} = \frac{1}{2} \]
- Solving for \( y \):
\[ y = 1 \]

### Final Answer:
Thus, the value of \( y \) is \( 1 \) unit.

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