Find the value of
`5(sin100^(@)cos27^(@)+sin27^(@)cos100^(@)).`

Answer

To solve the problem \( 5(\sin 100^\circ \cos 27^\circ + \sin 27^\circ \cos 100^\circ) \), we can use the sine addition formula. Here’s a step-by-step breakdown of the solution:

### Step 1: Identify the Sine Addition Formula
The sine addition formula states that:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]
In our case, we can let \( A = 100^\circ \) and \( B = 27^\circ \).

### Step 2: Apply the Sine Addition Formula
Using the sine addition formula, we can rewrite the expression:
\[ \sin 100^\circ \cos 27^\circ + \sin 27^\circ \cos 100^\circ = \sin(100^\circ + 27^\circ) \]
Thus, we have:
\[ \sin 100^\circ \cos 27^\circ + \sin 27^\circ \cos 100^\circ = \sin 127^\circ \]

### Step 3: Substitute Back into the Original Expression
Now, substituting back into the original expression gives us:
\[ 5(\sin 100^\circ \cos 27^\circ + \sin 27^\circ \cos 100^\circ) = 5 \sin 127^\circ \]

### Step 4: Simplify \( \sin 127^\circ \)
We can simplify \( \sin 127^\circ \) using the identity \( \sin(180^\circ - x) = \sin x \):
\[ \sin 127^\circ = \sin(180^\circ - 53^\circ) = \sin 53^\circ \]

### Step 5: Find the Value of \( \sin 53^\circ \)
From trigonometric tables or a calculator, we know:
\[ \sin 53^\circ = \frac{4}{5} \]

### Step 6: Calculate the Final Value
Now substituting this value back into our expression:
\[ 5 \sin 127^\circ = 5 \sin 53^\circ = 5 \times \frac{4}{5} = 4 \]

### Final Answer
Thus, the value of \( 5(\sin 100^\circ \cos 27^\circ + \sin 27^\circ \cos 100^\circ) \) is:
\[ \boxed{4} \]