Find the exact value of the expression. $$ \sin \frac{\pi}{6} \cos \frac{2 \pi}{3}-\cos \frac{\pi}{6} \sin \frac{2 \pi}{3} $$ $$ \sin \frac{\pi}{6} \cos \frac{2 \pi}{3}-\cos \frac{\pi}{6} \sin \frac{2 \pi}{3}=\square $$ (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

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UpStudy AI · Accepted Answer

To evaluate the expression:

$$
\sin \frac{\pi}{6} \cos \frac{2\pi}{3} - \cos \frac{\pi}{6} \sin \frac{2\pi}{3}
$$

We can recognize that it fits the formula for the sine of a difference of angles:

$$
\sin(A - B) = \sin A \cos B - \cos A \sin B
$$

Here, $$ A = \frac{\pi}{6} $$ and $$ B = \frac{2\pi}{3} $$. Applying the formula:

$$
\sin \left( \frac{\pi}{6} - \frac{2\pi}{3} \right) = \sin \left( -\frac{\pi}{2} \right)
$$

We know that:

$$
\sin \left( -\frac{\pi}{2} \right) = -1
$$

Alternatively, performing the calculations directly:

1. Calculate each trigonometric function:
   - $$ \sin \frac{\pi}{6} = \frac{1}{2} $$
   - $$ \cos \frac{2\pi}{3} = \cos 120^\circ = -\frac{1}{2} $$
   - $$ \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} $$
   - $$ \sin \frac{2\pi}{3} = \sin 120^\circ = \frac{\sqrt{3}}{2} $$

2. Substitute these values into the expression:
   $$
   \left(\frac{1}{2}\right) \left(-\frac{1}{2}\right) - \left(\frac{\sqrt{3}}{2}\right) \left(\frac{\sqrt{3}}{2}\right) = -\frac{1}{4} - \frac{3}{4} = -1
   $$

**Final Answer:** $$-1$$
$$-1$$

Find the exact value of the expression.

(Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)

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Find the exact value of the expression. (Simplify your answer, including any radicals. Use integers or fractions for any numbers in the expression.)