In a triangle `ABC,angleABC=75^(@)andangleACB=(pi^(c ))/(4)` . The circular measure of `angleBAC` is

Options

A. `(5 pi)/(12)` radian
B. `(pi)/(3)` radian
C. `(pi)/(6)` radian
D. `(pi)/(2)` radian

Answer

To find the circular measure of angle \( BAC \) in triangle \( ABC \), where \( \angle ABC = 75^\circ \) and \( \angle ACB = \frac{\pi}{4} \), we will follow these steps:

### Step 1: Convert angle \( ACB \) from radians to degrees
We know that:
\[ \frac{\pi}{4} \text{ radians} = \frac{\pi}{4} \times \frac{180^\circ}{\pi} = 45^\circ \]

### Step 2: Use the triangle angle sum property
The sum of the angles in a triangle is \( 180^\circ \). Therefore, we can write:
\[ \angle A + \angle B + \angle C = 180^\circ \]
Substituting the known values:
\[ \angle A + 75^\circ + 45^\circ = 180^\circ \]

### Step 3: Simplify the equation
Combine the known angles:
\[ \angle A + 120^\circ = 180^\circ \]

### Step 4: Solve for angle \( A \)
Subtract \( 120^\circ \) from both sides:
\[ \angle A = 180^\circ - 120^\circ = 60^\circ \]

### Step 5: Convert angle \( A \) from degrees to radians
To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180^\circ} \):
\[ \angle A = 60^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{3} \text{ radians} \]

### Final Answer
Thus, the circular measure of angle \( BAC \) is:
\[ \frac{\pi}{3} \text{ radians} \]

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