Answer :
To determine the value of [tex]\(\tan(60^\circ)\)[/tex], let's review some trigonometry concepts. The tangent of an angle in a right triangle is the ratio of the length of the opposite side to the length of the adjacent side.
For the specific angle of [tex]\(60^\circ\)[/tex], we can use our knowledge of special triangles, particularly the 30-60-90 triangle. In a 30-60-90 triangle, the ratios of the sides are as follows:
- The side opposite the 30° angle is [tex]\( \frac{1}{2} \)[/tex] of the hypotenuse.
- The side opposite the 60° angle is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] of the hypotenuse.
- The hypotenuse is the longest side.
So, if we denote:
- Opposite to [tex]\(60^\circ\)[/tex] as [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
- Adjacent to [tex]\(60^\circ\)[/tex] as [tex]\( \frac{1}{2} \)[/tex]
Then the tangent of [tex]\(60^\circ\)[/tex] is given by:
[tex]\[ \tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3} \][/tex]
So, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \sqrt{3} \][/tex]
For the specific angle of [tex]\(60^\circ\)[/tex], we can use our knowledge of special triangles, particularly the 30-60-90 triangle. In a 30-60-90 triangle, the ratios of the sides are as follows:
- The side opposite the 30° angle is [tex]\( \frac{1}{2} \)[/tex] of the hypotenuse.
- The side opposite the 60° angle is [tex]\( \frac{\sqrt{3}}{2} \)[/tex] of the hypotenuse.
- The hypotenuse is the longest side.
So, if we denote:
- Opposite to [tex]\(60^\circ\)[/tex] as [tex]\( \frac{\sqrt{3}}{2} \)[/tex]
- Adjacent to [tex]\(60^\circ\)[/tex] as [tex]\( \frac{1}{2} \)[/tex]
Then the tangent of [tex]\(60^\circ\)[/tex] is given by:
[tex]\[ \tan(60^\circ) = \frac{\text{opposite}}{\text{adjacent}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \frac{\sqrt{3}}{2} \times \frac{2}{1} = \sqrt{3} \][/tex]
So, the value of [tex]\(\tan(60^\circ)\)[/tex] is [tex]\(\sqrt{3}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \sqrt{3} \][/tex]
