Question

write the correct equation for this problem. do not solve. image of a right triangle with one angle 58°, hypotenuse? wait, no, the right angle, one leg 11, another leg x, and the angle opposite to x? wait, the angle is 58°, right triangle, right angle, side labeled 11, side x, angle 58°

Explanation:

Step1: Identify triangle type

It's a right - triangle with an angle of \(58^{\circ}\), hypotenuse (opposite right - angle) is not given, the side of length 11 is opposite the \(58^{\circ}\) angle? Wait, no. Wait, the right - angle, the angle of \(58^{\circ}\), the side adjacent to \(58^{\circ}\) is \(x\), and the side opposite to \(58^{\circ}\) is 11? Wait, no, let's recall trigonometric ratios. In a right - triangle, \(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\), \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). Wait, the angle is \(58^{\circ}\), the side with length 11: let's see, the right - angle, so the two legs are \(x\) and 11? Wait, no, the angle of \(58^{\circ}\), so if we take \(\tan(58^{\circ})=\frac{11}{x}\)? Wait, no, maybe \(\tan(58^{\circ})=\frac{\text{opposite}}{\text{adjacent}}\). If the angle is \(58^{\circ}\), the side opposite to \(58^{\circ}\) is 11, and the side adjacent is \(x\), then \(\tan(58^{\circ})=\frac{11}{x}\). Or if the side adjacent is 11 and opposite is \(x\), then \(\tan(58^{\circ})=\frac{x}{11}\). Wait, let's check the triangle. The right - angle, the angle of \(58^{\circ}\), so the sides: the side labeled 11 and \(x\) are the two legs. The angle of \(58^{\circ}\): the side opposite to \(58^{\circ}\) is 11, and the side adjacent is \(x\). So \(\tan(58^{\circ})=\frac{11}{x}\) or \(\tan(58^{\circ})=\frac{\text{opposite}}{\text{adjacent}}=\frac{11}{x}\). Alternatively, if we consider \(\cot(58^{\circ})=\frac{x}{11}\), but \(\cot\theta=\frac{1}{\tan\theta}\). So the equation can be \(\tan(58^{\circ})=\frac{11}{x}\) or \(x = \frac{11}{\tan(58^{\circ})}\) (but the question says write the correct equation, not solve). Wait, let's re - examine. The angle is \(58^{\circ}\), the side with length 11: if we take the angle \(58^{\circ}\), the side adjacent to \(58^{\circ}\) is \(x\), and the side opposite is 11. Then \(\tan(58^{\circ})=\frac{11}{x}\). So the equation is \(\tan(58^{\circ})=\frac{11}{x}\) (or \(x=\frac{11}{\tan(58^{\circ})}\) but the first form is the equation for the relationship).

Step2: Formulate the trigonometric equation

Using the tangent ratio in a right - triangle, where \(\theta = 58^{\circ}\), opposite side to \(\theta\) is 11, and adjacent side is \(x\). So \(\tan(58^{\circ})=\frac{11}{x}\)

Answer:

\(\tan(58^{\circ})=\frac{11}{x}\)