Question

determining trigonometric ratios
consider △dfe. what are the inputs or outputs of the following trigonometric ratios? express the ratios in simplest terms.
$sin(square)=\frac{4}{5}$
$cos(f)=square$
$\tan(d)=square$

Explanation:

Step1: Find missing side (already given)

We confirm the right triangle $\triangle DFE$ with right angle at $E$: $DE=20$, $EF=15$, $DF=25$.

Step2: Solve $\sin(\square)=\frac{4}{5}$

Recall $\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. $\frac{20}{25}=\frac{4}{5}$, so the side opposite the angle is $DE=20$, which is opposite $\angle F$.

Step3: Calculate $\cos(F)$

$\cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}}$. For $\angle F$, adjacent side is $EF=15$, hypotenuse $DF=25$.
$\cos(F)=\frac{15}{25}=\frac{3}{5}$

Step4: Calculate $\tan(D)$

$\tan(\theta)=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle D$, opposite side is $EF=15$, adjacent side is $DE=20$.
$\tan(D)=\frac{15}{20}=\frac{3}{4}$

Answer:

$\sin(\boldsymbol{F}) = \frac{4}{5}$
$\cos(F) = \boldsymbol{\frac{3}{5}}$
$\tan(D) = \boldsymbol{\frac{3}{4}}$