Answer :
To determine the value of \(\tan 30^\circ\), we begin by using the definition of the tangent function in terms of sine and cosine. The tangent of an angle in a right triangle is given by the ratio of the opposite side to the adjacent side. For an angle \(\theta\), it can also be expressed using the sine and cosine functions:
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
For the angle \(30^\circ\), we need the values of \(\sin 30^\circ\) and \(\cos 30^\circ\). Using the unit circle or known values of sine and cosine for common angles, we have:
[tex]\[ \sin 30^\circ = \frac{1}{2} \][/tex]
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \][/tex]
Now, substituting these values into the tangent formula, we get:
[tex]\[ \tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} \][/tex]
To simplify this expression, we divide \(\frac{1}{2}\) by \(\frac{\sqrt{3}}{2}\). Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[ \tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{\sqrt{3}} = \frac{1 \times 2}{2 \times \sqrt{3}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} \][/tex]
To confirm, converting \(\frac{1}{\sqrt{3}}\) into decimal form yields approximately:
[tex]\[ \frac{1}{\sqrt{3}} \approx 0.5773502691896258 \][/tex]
Therefore, the value of \(\tan 30^\circ\) is \(\frac{1}{\sqrt{3}}\), and the correct answer is:
[tex]\[ \boxed{\frac{1}{\sqrt{3}}} \][/tex]
[tex]\[ \tan \theta = \frac{\sin \theta}{\cos \theta} \][/tex]
For the angle \(30^\circ\), we need the values of \(\sin 30^\circ\) and \(\cos 30^\circ\). Using the unit circle or known values of sine and cosine for common angles, we have:
[tex]\[ \sin 30^\circ = \frac{1}{2} \][/tex]
[tex]\[ \cos 30^\circ = \frac{\sqrt{3}}{2} \][/tex]
Now, substituting these values into the tangent formula, we get:
[tex]\[ \tan 30^\circ = \frac{\sin 30^\circ}{\cos 30^\circ} = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} \][/tex]
To simplify this expression, we divide \(\frac{1}{2}\) by \(\frac{\sqrt{3}}{2}\). Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[ \tan 30^\circ = \frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}} = \frac{1}{2} \times \frac{2}{\sqrt{3}} = \frac{1 \times 2}{2 \times \sqrt{3}} = \frac{2}{2\sqrt{3}} = \frac{1}{\sqrt{3}} \][/tex]
To confirm, converting \(\frac{1}{\sqrt{3}}\) into decimal form yields approximately:
[tex]\[ \frac{1}{\sqrt{3}} \approx 0.5773502691896258 \][/tex]
Therefore, the value of \(\tan 30^\circ\) is \(\frac{1}{\sqrt{3}}\), and the correct answer is:
[tex]\[ \boxed{\frac{1}{\sqrt{3}}} \][/tex]
