Question

57 which equation is correct? (a) right triangle with right angle at b, angle at c is 26 degrees, hypotenuse ac is 14 ft, ab is x, bc is y a \\( \sin(26) = \frac{x}{14} \\) b \\( \sin(26) = \frac{14}{x} \\) c \\( \cos(26) = \frac{x}{14} \\) d \\( \cos(26) = \frac{14}{x} \\)

Explanation:

Step1: Recall SOHCAHTOA

In a right triangle, \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}\), \(\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}\).

Step2: Identify sides for \(\theta = 26^\circ\)

• Hypotenuse: \(14\) ft (side opposite right angle, \(AC\)).
• Opposite side to \(26^\circ\): \(x\) (side \(AB\), opposite \(\angle C\)).
• Adjacent side to \(26^\circ\): \(y\) (side \(BC\), adjacent to \(\angle C\)).

Step3: Apply \(\sin\) formula

Using \(\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}\), substitute \(\theta = 26^\circ\), opposite \(= x\), hypotenuse \(= 14\):
\(\sin(26^\circ)=\frac{x}{14}\).

Check other options:

• Option B: \(\sin(26^\circ)=\frac{14}{x}\) is wrong (opposite and hypotenuse reversed).
• Option C: \(\cos(26^\circ)=\frac{x}{14}\) is wrong (uses opposite for cosine, which needs adjacent).
• Option D: \(\cos(26^\circ)=\frac{14}{x}\) is wrong (adjacent and hypotenuse reversed, and wrong side for cosine).

Answer:

A. \(\sin(26)=\frac{x}{14}\)