In a `triangleABC`, if `sinA - cosB = cos C`, then the measured of `angleB` is

Options

A. `pi/2`
B. `pi/3`
C. `pi/4`
D. `pi/6`

Answer

To solve the problem, we start with the equation given in the triangle \( ABC \):

\[ \sin A - \cos B = \cos C \]

### Step 1: Rearranging the Equation
We can rearrange the equation to isolate \( \sin A \):

\[ \sin A = \cos B + \cos C \]

### Step 2: Using the Cosine Rule
In triangle \( ABC \), we know that:

\[ C = 180^\circ - A - B \]

Thus, we can express \( \cos C \) using the cosine of the sum of angles:

\[ \cos C = \cos(180^\circ - A - B) = -\cos(A + B) \]

### Step 3: Applying the Cosine Addition Formula
Using the cosine addition formula, we have:

\[ \cos(A + B) = \cos A \cos B - \sin A \sin B \]

So, we can rewrite \( \cos C \):

\[ \cos C = -(\cos A \cos B - \sin A \sin B) \]

### Step 4: Substitute Back into the Equation
Substituting this back into our rearranged equation gives:

\[ \sin A = \cos B - (\cos A \cos B - \sin A \sin B) \]

### Step 5: Simplifying the Equation
This simplifies to:

\[ \sin A + \cos A \cos B = \cos B + \sin A \sin B \]

Rearranging gives:

\[ \sin A (1 - \sin B) = \cos B (1 - \cos A) \]

### Step 6: Analyzing the Angles
Since we are looking for the measure of angle \( B \), we can analyze the case when \( B = 90^\circ \):

If \( B = 90^\circ \), then:

\[ \cos B = 0 \quad \text{and} \quad \sin B = 1 \]

Substituting \( B = 90^\circ \) into the original equation:

\[ \sin A - 0 = \cos C \]

This implies:

\[ \sin A = \cos C \]

### Step 7: Conclusion
Since \( C = 180^\circ - A - 90^\circ \), we have:

\[ C = 90^\circ - A \]

Thus, \( \cos C = \sin A \), confirming our assumption. Therefore, the measure of angle \( B \) is:

\[ \boxed{90^\circ} \]