Question

given right triangle xyz, what is the value of tan(60°)? triangle diagram: right-angled at z, ∠x=60°, ∠y=30°, xz=21, xy=42; options: 1/2, √3, √3/2, 2/1

Explanation:

Step1: Recall the definition of tangent

In a right triangle, the tangent of an angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \(60^\circ\) (angle at \(X\)) in right triangle \(XYZ\) (right - angled at \(Z\)):

• The side opposite to \(60^\circ\) is \(YZ\). From the triangle, we can also use the 30 - 60 - 90 triangle ratios. In a 30 - 60 - 90 triangle, the sides are in the ratio \(1:\sqrt{3}:2\) (opposite to \(30^\circ\), \(60^\circ\), and hypotenuse respectively). Or we can use the given side lengths. The side adjacent to \(60^\circ\) (angle at \(X\)) is \(XZ = 21\), and the side opposite to \(60^\circ\) is \(YZ\). We know that in a right triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\).
• Alternatively, we can use the formula for \(\tan(60^\circ)\) from the trigonometric ratios. We know that \(\tan(60^\circ)=\sqrt{3}\) (from the unit circle or trigonometric table, where for \(60^\circ\), \(\sin(60^\circ)=\frac{\sqrt{3}}{2}\) and \(\cos(60^\circ)=\frac{1}{2}\), and \(\tan\theta = \frac{\sin\theta}{\cos\theta}\), so \(\tan(60^\circ)=\frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}}=\sqrt{3}\)).
• Let's verify with the triangle sides. The length of \(XZ = 21\), and if we consider angle \(X = 60^\circ\), the side opposite to \(60^\circ\) ( \(YZ\)): In a 30 - 60 - 90 triangle, the hypotenuse \(XY = 42\), the side opposite to \(30^\circ\) (angle at \(Y\)) is \(XZ = 21\) (since in 30 - 60 - 90 triangle, the side opposite to \(30^\circ\) is half of the hypotenuse, \(42\div2 = 21\)), and the side opposite to \(60^\circ\) is \(XZ\times\sqrt{3}=21\sqrt{3}\). Then \(\tan(60^\circ)=\frac{\text{opposite}}{\text{adjacent}}=\frac{21\sqrt{3}}{21}=\sqrt{3}\).

Answer:

\(\sqrt{3}\)