In a right triangle ABC, right angled at C, if a = 7 cm and b = `7 sqrt(3)` cm, then `angleA`=

Options

A. `30^(@)`
B. `60^(@)`
C. `45^(@)`
D. none of these

Explanation

To find the angle A in triangle ABC, where C is the right angle, we can use trigonometric ratios. Here’s a step-by-step solution:

### Step 1: Understand the triangle
In triangle ABC, we know:
- Angle C = 90 degrees (right angle)
- Side a (opposite to angle A) = 7 cm
- Side b (opposite to angle B) = 7√3 cm

### Step 2: Identify the sides
In a right triangle:
- The side opposite angle A is denoted as a.
- The side opposite angle B is denoted as b.
- The hypotenuse (side opposite the right angle) is denoted as c.

### Step 3: Use the tangent function
To find angle A, we can use the tangent function:
\[ \tan A = \frac{\text{Opposite}}{\text{Adjacent}} = \frac{a}{b} \]
Here, a = 7 cm (opposite to angle A) and b = 7√3 cm (adjacent to angle A).

### Step 4: Substitute the values
Substituting the values into the tangent function:
\[ \tan A = \frac{7}{7\sqrt{3}} \]
This simplifies to:
\[ \tan A = \frac{1}{\sqrt{3}} \]

### Step 5: Find angle A using the inverse tangent
To find angle A, we take the inverse tangent:
\[ A = \tan^{-1}\left(\frac{1}{\sqrt{3}}\right) \]

### Step 6: Determine the angle
From trigonometric values, we know:
\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \]
Thus:
\[ A = 30^\circ \]

### Conclusion
The measure of angle A is:
\[ \boxed{30^\circ} \]

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