ABC is a right angled triangle, right angled at C. If `angleA = 60^(@) and AB = 40` unit, then the length of AC is

Options

A. 30 unit
B. 20 unit
C. 40 unit
D. None of these

Answer

To find the length of AC in triangle ABC, where angle A is 60 degrees and AB (the hypotenuse) is 40 units, we can use trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Identify the triangle and known values
We have a right-angled triangle ABC with:
- Angle A = 60 degrees
- AB (hypotenuse) = 40 units
- Angle C = 90 degrees (right angle)

### Step 2: Use the cosine function
In a right triangle, the cosine of an angle is defined as the ratio of the length of the adjacent side to the hypotenuse. Here, we can use:
\[ \cos(A) = \frac{\text{Adjacent (AC)}}{\text{Hypotenuse (AB)}} \]

### Step 3: Substitute known values into the cosine formula
We know:
- \(\cos(60^\circ) = \frac{1}{2}\)
- Hypotenuse (AB) = 40 units

So we can write:
\[ \cos(60^\circ) = \frac{AC}{40} \]
Substituting the value of \(\cos(60^\circ)\):
\[ \frac{1}{2} = \frac{AC}{40} \]

### Step 4: Solve for AC
To find AC, we can rearrange the equation:
\[ AC = 40 \cdot \frac{1}{2} \]
\[ AC = 20 \text{ units} \]

### Conclusion
The length of AC is 20 units.

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