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Find the exact value of [tex]\(\tan 30^{\circ} \times \sin 60^{\circ}\)[/tex]. Give your answer in its simplest form.

Sagot :

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]