Question

express the trig ratios as fractions in simplest terms.

answer

cos e =

sin f =

cos e and sin f are equal because \\(\overline{de}\\) is \\(\angle e\\) and \\(\angle f\\).

Explanation:

Step1: Recall cosine and sine definitions

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

Step2: Find \(\cos E\)

For \(\angle E\), adjacent side is \(DE = 15\), hypotenuse is \(FE=39\). So \(\cos E=\frac{DE}{FE}=\frac{15}{39}\). Simplify: \(\frac{15\div3}{39\div3}=\frac{5}{13}\).

Step3: Find \(\sin F\)

For \(\angle F\), opposite side is \(DE = 15\), hypotenuse is \(FE = 39\). So \(\sin F=\frac{DE}{FE}=\frac{15}{39}=\frac{5}{13}\).

Answer:

\(\cos E=\frac{5}{13}\)
\(\sin F=\frac{5}{13}\)