Triangle ABC is a right angled traingle at B. If angle A= 60degree, then what is the value of Cot C?

Options

A. a. √2
B. b. 1/√3
C. c. √3
D. d. none of these

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Triangle
We have a right-angled triangle ABC with the right angle at B. We know that angle A = 60 degrees.

### Step 2: Find Angle C
In any triangle, the sum of the angles is always 180 degrees. Therefore, we can write the equation for the angles in triangle ABC as follows:

\[ \text{Angle A} + \text{Angle B} + \text{Angle C} = 180^\circ \]

Given:
- Angle A = 60 degrees
- Angle B = 90 degrees (since it is a right-angled triangle)

Substituting the known values into the equation:

\[ 60^\circ + 90^\circ + \text{Angle C} = 180^\circ \]

### Step 3: Solve for Angle C
Now, we can solve for Angle C:

\[ \text{Angle C} = 180^\circ - 60^\circ - 90^\circ \]
\[ \text{Angle C} = 180^\circ - 150^\circ \]
\[ \text{Angle C} = 30^\circ \]

### Step 4: Find Cotangent of Angle C
We need to find the value of cotangent of angle C, which is:

\[ \cot C = \cot(30^\circ) \]

### Step 5: Use the Cotangent Formula
The cotangent of an angle is the reciprocal of the tangent of that angle. Therefore:

\[ \cot(30^\circ) = \frac{1}{\tan(30^\circ)} \]

We know that:

\[ \tan(30^\circ) = \frac{1}{\sqrt{3}} \]

Thus:

\[ \cot(30^\circ) = \frac{1}{\frac{1}{\sqrt{3}}} = \sqrt{3} \]

### Conclusion
The value of \(\cot C\) is \(\sqrt{3}\).

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