To find the exact value of [tex]\(\tan 30^\circ \times \sin 60^\circ\)[/tex], let's break it down step by step.
1. Determine [tex]\(\tan 30^\circ\)[/tex]:
The tangent of 30 degrees is a well-known trigonometric value:
[tex]\[
\tan 30^\circ = \frac{1}{\sqrt{3}}
\][/tex]
2. Determine [tex]\(\sin 60^\circ\)[/tex]:
The sine of 60 degrees is also a fundamental trigonometric value:
[tex]\[
\sin 60^\circ = \frac{\sqrt{3}}{2}
\][/tex]
3. Multiply these values together:
Now, multiply [tex]\(\tan 30^\circ\)[/tex] by [tex]\(\sin 60^\circ\)[/tex]:
[tex]\[
\tan 30^\circ \times \sin 60^\circ = \left( \frac{1}{\sqrt{3}} \right) \times \left( \frac{\sqrt{3}}{2} \right)
\][/tex]
4. Simplify the expression:
Simplify the product:
[tex]\[
\frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{2} = \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot 2} = \frac{\sqrt{3}}{2\sqrt{3}} = \frac{1}{2}
\][/tex]
Therefore, the exact value of [tex]\(\tan 30^\circ \times \sin 60^\circ\)[/tex] is:
[tex]\[
\boxed{\frac{1}{2}}
\][/tex]
