What is the value of `|(sin 10^(@)-cos 10^(@)),(sin 80^(@),cos 80^(@))|` ?

Options

A. 0
B. 1
C. -1
D. `(1)/(2)`

Answer

To find the value of the determinant

\[ D = \begin{vmatrix} \sin 10^\circ & -\cos 10^\circ \\ \sin 80^\circ & \cos 80^\circ \end{vmatrix}, \]

we will follow these steps:

### Step 1: Write the determinant formula
The value of a 2x2 determinant is given by the formula:

\[ D = ad - bc, \]

where \( a, b, c, d \) are the elements of the matrix:

\[ D = \begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc. \]

### Step 2: Identify the elements of the determinant
In our case, we have:

- \( a = \sin 10^\circ \)
- \( b = -\cos 10^\circ \)
- \( c = \sin 80^\circ \)
- \( d = \cos 80^\circ \)

### Step 3: Substitute the values into the determinant formula
Now, substituting these values into the determinant formula, we get:

\[ D = (\sin 10^\circ)(\cos 80^\circ) - (-\cos 10^\circ)(\sin 80^\circ). \]

### Step 4: Simplify the expression
This simplifies to:

\[ D = \sin 10^\circ \cos 80^\circ + \cos 10^\circ \sin 80^\circ. \]

### Step 5: Use the sine addition formula
We can recognize that this expression is the sine of a sum:

\[ D = \sin(10^\circ + 80^\circ) = \sin 90^\circ. \]

### Step 6: Calculate the value of sine
Since \( \sin 90^\circ = 1 \), we find that:

\[ D = 1. \]

### Final Answer
Thus, the value of the determinant is:

\[ \boxed{1}. \]