Question

determining trigonometric ratios. 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 sine ratio formula

$\sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}}$. Given $\sin(\theta)=\frac{4}{5}$, in right - triangle $\triangle DFE$, if we assume the opposite side to the angle is $4x$ and hypotenuse is $5x$. The sides of the triangle are $15,20,25$ (a $3 - 4-5$ multiple triangle). For $\sin(\theta)=\frac{4}{5}$, the opposite side is $20$ and hypotenuse is $25$, so the angle is $\angle F$.

Step2: Recall cosine ratio formula

$\cos(F)=\frac{\text{adjacent}}{\text{hypotenuse}}$. For $\angle F$, the adjacent side to $\angle F$ is $15$ and the hypotenuse is $25$. So $\cos(F)=\frac{15}{25}=\frac{3}{5}$.

Step3: Recall tangent ratio formula

$\tan(D)=\frac{\text{opposite}}{\text{adjacent}}$. For $\angle D$, the opposite side is $15$ and the adjacent side is $20$. So $\tan(D)=\frac{15}{20}=\frac{3}{4}$.

Answer:

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