Question

write the equation you would set up for this problem. do not solve.

Explanation:

Step1: Identify the triangle type

The triangle is a right - triangle with hypotenuse \( c = 14 \), an acute angle \( \theta=37^{\circ} \), and the adjacent side to the angle \( 37^{\circ} \) is \( x \).

Step2: Recall the cosine function definition

In a right - triangle, the cosine of an acute angle \( \theta \) is defined as \( \cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}} \).
For this triangle, \( \theta = 37^{\circ} \), adjacent side \( = x \), and hypotenuse \( = 14 \). So we can set up the equation using the cosine function: \( \cos(37^{\circ})=\frac{x}{14} \) (or we can also use the sine function. If we consider the angle and the opposite side, but in this case, since \( x \) is adjacent to \( 37^{\circ} \), cosine is more appropriate. Another way: If we look at the angle \( 37^{\circ} \), the adjacent side is \( x \) and hypotenuse is 14, so \( \cos(37^{\circ})=\frac{x}{14} \) or we can also write \( x = 14\cos(37^{\circ}) \) as the equation to set up, but the key is to use the trigonometric ratio. Also, we can use the sine function if we consider the other angle, but the most direct for \( x \) (adjacent to \( 37^{\circ} \)) is cosine. So the equation is \( \cos(37^{\circ})=\frac{x}{14} \) (or \( x = 14\cos37^{\circ} \))

Answer:

\(\cos(37^{\circ})=\frac{x}{14}\) (or \(x = 14\cos37^{\circ}\))