Question

consider △dfe. what are the inputs or outputs of the following trigonometric ratios? express the ratios in simplest terms. sin( ) = 4/5, cos(f) = <>, tan(d) = <>

Explanation:

Step1: Recall tangent formula

Tangent of an angle in a right - triangle is opposite/adjacent. For $\angle D$, the opposite side to $\angle D$ is $EF = 15$ and the adjacent side is $DE=20$. So, $\tan(D)=\frac{EF}{DE}$.
$\tan(D)=\frac{15}{20}=\frac{3}{4}$

Step2: Recall cosine formula

Cosine of an angle in a right - triangle is adjacent/hypotenuse. For $\angle F$, the adjacent side to $\angle F$ is $EF = 15$ and the hypotenuse is $DF = 25$. So, $\cos(F)=\frac{EF}{DF}$.
$\cos(F)=\frac{15}{25}=\frac{3}{5}$

Step3: Determine the angle for $\sin$ value

We know that $\sin(\theta)=\frac{4}{5}$. In right - triangle $DFE$, if $\sin(\theta)=\frac{4}{5}$, and $\sin$ is opposite/hypotenuse, the opposite side is $16$ (using the Pythagorean triple $3 - 4-5$ scaled up, or $a^{2}+b^{2}=c^{2}$, if $c = 20$ and one side is $12$, the other is $16$). The angle for which $\sin$ is $\frac{4}{5}$ is $\angle D$.

Answer:

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