Question

consider this right triangle. determine whether each equation is correct. select yes or no for each equation.

$\tan(d)=\frac{5}{12}$

$\sin(f)=\frac{12}{13}$

$\cos(f)=\frac{13}{5}$

$\sin(d)=\frac{13}{5}$

Explanation:

Step1: Recall trigonometric ratios

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

Step2: Analyze \(\tan(D)\)

For \(\angle D\), the opposite side to \(\angle D\) is \(EF = 5\) and the adjacent side is \(DE=12\). So, \(\tan(D)=\frac{EF}{DE}=\frac{5}{12}\)

Step3: Analyze \(\sin(F)\)

For \(\angle F\), the opposite side is \(DE = 12\) and the hypotenuse is \(DF = 13\). So, \(\sin(F)=\frac{DE}{DF}=\frac{12}{13}\)

Step4: Analyze \(\cos(F)\)

For \(\angle F\), the adjacent side is \(EF = 5\) and the hypotenuse is \(DF = 13\). So, \(\cos(F)=\frac{EF}{DF}=\frac{5}{13}
eq\frac{13}{5}\)

Step5: Analyze \(\sin(D)\)

For \(\angle D\), the opposite side is \(EF = 5\) and the hypotenuse is \(DF = 13\). So, \(\sin(D)=\frac{EF}{DF}=\frac{5}{13}
eq\frac{13}{5}\)

Answer:

\(\tan(D)=\frac{5}{12}\): Yes
\(\sin(F)=\frac{12}{13}\): Yes
\(\cos(F)=\frac{13}{5}\): No
\(\sin(D)=\frac{13}{5}\): No