Question

solve for n. if needed, round to the nearest hundredth.

n = 12.98

n = 13.02

n = 3.70

n = 3.75

Explanation:

Step1: Use the cosine function

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 68^{\circ}\), the adjacent side to the \(68^{\circ}\) angle is \(n\), and the hypotenuse is \(14\). So, \(\cos(68^{\circ})=\frac{n}{14}\).

Step2: Solve for \(n\)

Multiply both sides of the equation \(\cos(68^{\circ})=\frac{n}{14}\) by \(14\). We get \(n = 14\times\cos(68^{\circ})\).
Since \(\cos(68^{\circ})\approx0.3746\), then \(n=14\times0.3746 = 5.2444\) (This is wrong. Wait, no! Wait, wrong angle - side relationship. Wait, in a right - triangle, if the angle is \(68^{\circ}\), and we want the side \(n\) (assuming the right - angle is opposite the hypotenuse). Wait, no, using the sine function. \(\sin(68^{\circ})=\frac{\text{opposite}}{\text{hypotenuse}}\) is wrong. Wait, no, if the angle is \(68^{\circ}\), and we have a right - triangle. Let's use \(\sin\alpha=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\alpha=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan\alpha=\frac{\text{opposite}}{\text{adjacent}}\). Wait, actually, using \(\sin(68^{\circ})=\frac{n}{14}\) (if \(n\) is the side opposite the non - right angle. Wait, no, in a right - triangle with hypotenuse \(14\), angle \(68^{\circ}\), and we want \(n\). Using \(\sin(68^{\circ})=\frac{n}{14}\), \(n = 14\times\sin(68^{\circ})\). Since \(\sin(68^{\circ})\approx0.9272\), \(n=14\times0.9272=12.9808\approx12.98\)

Answer:

\(n = 12.98\)