Question

multiple choice 1 point
give the exact value.
tan 60°
√3
√3/3
2
√3/2

multiple choice 1 point
give the exact value.
sin 60°
√3/3
1/2
√3/2

Explanation:

Step1: Recall the value of $\tan60^{\circ}$

We know from the unit - circle or special right - triangle (30 - 60 - 90 triangle) that in a 30 - 60 - 90 triangle with sides \(1,\sqrt{3},2\) (where the side opposite \(30^{\circ}\) is \(1\), the side opposite \(60^{\circ}\) is \(\sqrt{3}\) and the hypotenuse is \(2\)). The formula for \(\tan\theta=\frac{\sin\theta}{\cos\theta}\), and for \(\theta = 60^{\circ}\), \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\), \(\cos60^{\circ}=\frac{1}{2}\). Then \(\tan60^{\circ}=\frac{\sin60^{\circ}}{\cos60^{\circ}}=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}\).

Step2: Recall the value of $\sin60^{\circ}$

From the 30 - 60 - 90 triangle, \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \(\theta = 60^{\circ}\), the side opposite \(60^{\circ}\) is \(\sqrt{3}\) and the hypotenuse is \(2\). So \(\sin60^{\circ}=\frac{\sqrt{3}}{2}\).

Answer:

For \(\tan60^{\circ}\): \(\sqrt{3}\) (the first option).
For \(\sin60^{\circ}\): \(\frac{\sqrt{3}}{2}\) (the fourth option).