To determine which expression is equal to [tex]\(\left(\sin 60^{\circ}\right)\left(\cos 30^{\circ}\right) + \left(\cos 60^{\circ}\right)\left(\sin 30^{\circ}\right)\)[/tex], we need to recognize that this expression resembles a trigonometric identity.
This specific form matches the angle addition formula for sine:
[tex]\[
\sin(A + B) = \sin A \cos B + \cos A \sin B
\][/tex]
In our scenario:
- [tex]\(A = 60^{\circ}\)[/tex]
- [tex]\(B = 30^{\circ}\)[/tex]
Using the angle addition formula:
[tex]\[
\sin(60^{\circ} + 30^{\circ}) = \sin 60^{\circ} \cos 30^{\circ} + \cos 60^{\circ} \sin 30^{\circ}
\][/tex]
Now we need to verify the angle addition:
[tex]\[
60^{\circ} + 30^{\circ} = 90^{\circ}
\][/tex]
According to the sine function:
[tex]\[
\sin 90^{\circ} = 1
\][/tex]
Breaking it down with the given values:
- [tex]\(\sin 60^{\circ} = 0.8660254037844386\)[/tex]
- [tex]\(\cos 30^{\circ} = 0.8660254037844387\)[/tex]
- [tex]\(\cos 60^{\circ} = 0.5000000000000001\)[/tex]
- [tex]\(\sin 30^{\circ} = 0.49999999999999994\)[/tex]
Let's calculate the expression:
[tex]\[
(\sin 60^{\circ})(\cos 30^{\circ}) + (\cos 60^{\circ})(\sin 30^{\circ}) = (0.8660254037844386 \times 0.8660254037844387) + (0.5000000000000001 \times 0.49999999999999994) = 1
\][/tex]
This confirms that:
[tex]\[
\left( \sin 60^{\circ} \right) \left( \cos 30^{\circ} \right) + \left( \cos 60^{\circ} \right) \left( \sin 30^{\circ} \right) = \sin(60^{\circ} + 30^{\circ})
\][/tex]
Therefore, the expression inside the problem is precisely equal to:
[tex]\[
\sin \left( 60^{\circ} + 30^{\circ} \right)
\][/tex]
Hence, the correct answer is:
[tex]\[
\boxed{\text{J}} \quad \sin(60^{\circ} + 30^{\circ})
\][/tex]
