Question

1. find x.

image of a right triangle with one leg 12, angle 53°, and hypotenuse x
options: 15.92, 11.53, 16.45, 12.12

Explanation:

Step1: Identify trigonometric ratio

We have a right - triangle. The side with length 12 is adjacent to the \(53^{\circ}\) angle, and \(x\) is the hypotenuse. We use the cosine ratio, which is \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Here, \(\theta = 53^{\circ}\), adjacent \(= 12\), and hypotenuse \(=x\). So, \(\cos(53^{\circ})=\frac{12}{x}\).

Step2: Solve for \(x\)

We can re - arrange the formula to solve for \(x\). From \(\cos(53^{\circ})=\frac{12}{x}\), we get \(x=\frac{12}{\cos(53^{\circ})}\). We know that \(\cos(53^{\circ})\approx0.6018\). Then \(x = \frac{12}{0.6018}\approx19.94\)? Wait, no, maybe I mixed up opposite and adjacent. Wait, wait, the right angle is between \(x\) and 12? Wait, no, let's re - examine the triangle. Wait, the angle is \(53^{\circ}\), the side of length 12 is adjacent, and \(x\) is opposite? Wait, no, the right angle is marked. Let's look again. The right angle is at the vertex where \(x\) and 12 meet? Wait, no, the triangle has a right angle, and the angle of \(53^{\circ}\), side 12, and \(x\). Wait, maybe it's a tangent or cotangent? Wait, no, maybe I made a mistake. Wait, if the right angle is between the side of length 12 and the hypotenuse? No, let's start over.

Wait, in a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Let's assume that the angle of \(53^{\circ}\), the side adjacent to it is 12, and the side opposite to it is \(x\). Then \(\tan(53^{\circ})=\frac{x}{12}\). Since \(\tan(53^{\circ})\approx1.3270\), then \(x = 12\times\tan(53^{\circ})\approx12\times1.3270 = 15.924\approx15.92\). Ah, that makes sense. So I had the ratio wrong earlier. So the correct ratio is tangent, because \(x\) is opposite the \(53^{\circ}\) angle and 12 is adjacent. So \(\tan(53^{\circ})=\frac{x}{12}\), so \(x = 12\times\tan(53^{\circ})\).

Calculating \(\tan(53^{\circ})\approx1.327\), so \(x = 12\times1.327 = 15.924\approx15.92\).

Answer:

15.92 (the option with 15.92)