Question

consider right triangle \\( \triangle def \\) below.
which expressions are equivalent to \\( \cos ( \angle e ) \\)?

Explanation:

Step1: Recall the definition of cosine

In a right - triangle, \(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \(\angle E\), the adjacent side is \(DE\) and the hypotenuse is \(EF\). So, \(\cos(\angle E)=\frac{DE}{EF}\).

Step2: Use co - function identity

We know that in a right - triangle \(\triangle DEF\) with \(\angle D = 90^{\circ}\), \(\angle E+\angle F=90^{\circ}\). Also, \(\cos\alpha=\sin(90^{\circ}-\alpha)\). So, \(\cos(\angle E)=\sin(\angle F)\). For \(\angle F\), \(\sin(\angle F)=\frac{DE}{EF}\) (since \(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\) and the opposite side of \(\angle F\) is \(DE\) and hypotenuse is \(EF\)).

Answer:

\(\cos(\angle E)=\sin(\angle F)=\frac{DE}{EF}\)