Question

which is the correct set up to solve for x? cos 72 = 6/x tan 72 = 6/x sin 72 = 6/x

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$. Here, the angle is $72^{\circ}$, the side adjacent to the $72^{\circ}$ angle is 6, and the side opposite to the $72^{\circ}$ angle is $x$.

Step2: Select the correct trigonometric ratio

Since $\tan72^{\circ}=\frac{\text{opposite}}{\text{adjacent}}=\frac{x}{6}$, we can rewrite it as $x = 6\tan72^{\circ}$. The correct setup from the given options based on the ratio is $\tan72=\frac{x}{6}$ (equivalent to $x = 6\tan72$). But among the given options, if we consider the reciprocal relationship, we know that $\tan72=\frac{x}{6}$ can also be written as $\frac{1}{\tan72}=\frac{6}{x}$, and $\cot72=\frac{6}{x}$. However, if we assume a mis - writing and we consider the standard form of the tangent ratio, the correct setup should be $\tan72=\frac{x}{6}$. If we assume the options are written as they are, we know that $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$, so for $\theta = 72^{\circ}$, opposite is $x$ and adjacent is 6, and the correct equation should be $\tan72=\frac{x}{6}$ (none of the given options are correct in the standard form, but if we consider the reciprocal nature in a non - standard way of looking at the options, we note the error in the options). If we assume the options are just about the form of the trigonometric equation without considering the correct side - order strictly, we know that $\tan72=\frac{x}{6}$ is the correct relationship and among the given ones, we assume the closest correct one is based on the tangent ratio concept.

Answer:

None of the given options are correct in the standard form of the tangent ratio setup for this right - triangle. But if we consider the form of the equations in terms of the tangent relationship conceptually, the correct relationship for the right - triangle with opposite side $x$ and adjacent side 6 with respect to the $72^{\circ}$ angle is $\tan72=\frac{x}{6}$. If we assume the options are written as they are and we consider the tangent - related form, we note the error in the options' writing. If we had to choose the closest in terms of the trigonometric ratio concept, we would say that the correct ratio is $\tan72=\frac{x}{6}$ and the given options have a wrong side - order in the fraction for the tangent setup.