Find the value of
`sin^(2)(120^(@)-A)+sin^(2)A+sin^(2)(120^(@)+A):`

Options

A. `(2)/(3)`
B. `(3)/(2)`
C. `(5)/(2)`
D. `(sqrt3)/(2)`

Answer

To solve the problem, we need to find the value of the expression:

\[ \sin^2(120^\circ - A) + \sin^2 A + \sin^2(120^\circ + A) \]

### Step-by-Step Solution:

**Step 1: Use the sine angle subtraction and addition formulas.**

The sine of an angle can be expressed using the formulas:
- \(\sin(a - b) = \sin a \cos b - \cos a \sin b\)
- \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)

Applying these formulas:

\[ \sin(120^\circ - A) = \sin 120^\circ \cos A - \cos 120^\circ \sin A \]
\[ \sin(120^\circ + A) = \sin 120^\circ \cos A + \cos 120^\circ \sin A \]

**Step 2: Substitute the values of \(\sin 120^\circ\) and \(\cos 120^\circ\).**

We know:
- \(\sin 120^\circ = \sin(180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2}\)
- \(\cos 120^\circ = \cos(180^\circ - 60^\circ) = -\cos 60^\circ = -\frac{1}{2}\)

Substituting these values into the equations:

\[ \sin(120^\circ - A) = \frac{\sqrt{3}}{2} \cos A + \frac{1}{2} \sin A \]
\[ \sin(120^\circ + A) = \frac{\sqrt{3}}{2} \cos A - \frac{1}{2} \sin A \]

**Step 3: Square the sine terms.**

Now we calculate \(\sin^2(120^\circ - A)\) and \(\sin^2(120^\circ + A)\):

\[ \sin^2(120^\circ - A) = \left(\frac{\sqrt{3}}{2} \cos A + \frac{1}{2} \sin A\right)^2 \]
\[ = \frac{3}{4} \cos^2 A + \frac{\sqrt{3}}{2} \cos A \sin A + \frac{1}{4} \sin^2 A \]

\[ \sin^2(120^\circ + A) = \left(\frac{\sqrt{3}}{2} \cos A - \frac{1}{2} \sin A\right)^2 \]
\[ = \frac{3}{4} \cos^2 A - \frac{\sqrt{3}}{2} \cos A \sin A + \frac{1}{4} \sin^2 A \]

**Step 4: Combine all terms.**

Now, we add all three squared terms:

\[ \sin^2(120^\circ - A) + \sin^2 A + \sin^2(120^\circ + A) \]

Substituting the squared values:

\[ = \left(\frac{3}{4} \cos^2 A + \frac{\sqrt{3}}{2} \cos A \sin A + \frac{1}{4} \sin^2 A\right) + \sin^2 A + \left(\frac{3}{4} \cos^2 A - \frac{\sqrt{3}}{2} \cos A \sin A + \frac{1}{4} \sin^2 A\right) \]

Combining like terms:

\[ = \frac{3}{4} \cos^2 A + \frac{3}{4} \cos^2 A + \frac{1}{4} \sin^2 A + \frac{1}{4} \sin^2 A + \left(\frac{\sqrt{3}}{2} \cos A \sin A - \frac{\sqrt{3}}{2} \cos A \sin A\right) \]

This simplifies to:

\[ = \frac{3}{2} \cos^2 A + \frac{1}{2} \sin^2 A \]

**Step 5: Use the Pythagorean identity.**

Using the identity \(\cos^2 A + \sin^2 A = 1\):

\[ = \frac{3}{2} \cos^2 A + \frac{1}{2} (1 - \cos^2 A) \]
\[ = \frac{3}{2} \cos^2 A + \frac{1}{2} - \frac{1}{2} \cos^2 A \]
\[ = \frac{2}{2} \cos^2 A + \frac{1}{2} = \cos^2 A + \frac{1}{2} \]

**Final Step: Conclusion.**

Thus, the value of the expression is:

\[ \frac{3}{2} \]