To find the exact value of [tex]\(\tan 30^\circ\)[/tex], we can use the known values for the sine and cosine of 30 degrees.
First, recall that:
[tex]\[ \sin 30^\circ = \frac{1}{2} \][/tex]
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \][/tex]
Now, the tangent of an angle is defined as the ratio of the sine of the angle to the cosine of the angle:
[tex]\[ \tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} \][/tex]
Substitute the known values for [tex]\(\sin 30^\circ\)[/tex] and [tex]\(\cos 30^\circ\)[/tex]:
[tex]\[ \tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} \][/tex]
To simplify this expression, divide the numerator by the denominator:
[tex]\[ \tan 30^\circ = \frac{1/2}{\sqrt{3}/2} \][/tex]
When dividing fractions, multiply by the reciprocal of the denominator:
[tex]\[ \tan 30^\circ = \frac{1/2} {\sqrt{3}/2} = \left(\frac{1}{2}\right) \times \left(\frac{2}{\sqrt{3}}\right) \][/tex]
The [tex]\(2\)[/tex]s in the numerator and denominator cancel each other out:
[tex]\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \][/tex]
To express this fraction in simplest form, rationalize the denominator by multiplying both the numerator and the denominator by [tex]\(\sqrt{3}\)[/tex]:
[tex]\[ \tan 30^\circ = \frac{1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \][/tex]
Thus, the exact value of [tex]\(\tan 30^\circ\)[/tex] is:
[tex]\[ \frac{\sqrt{3}}{3} \][/tex]
