Question

which trigonometric ratios are correct for triangle def? select three options. sin(d) = 24/25 cos(e) = 7/25 tan(d) = 24/7 sin(e) = 7/25 tan(d) = 7/24

Explanation:

Step1: Recall trigonometric - ratio definitions

In a right - triangle, $\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}$, $\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}$, $\tan\theta=\frac{\text{opposite}}{\text{adjacent}}$. First, find the hypotenuse of right - triangle $DEF$ using the Pythagorean theorem $c=\sqrt{a^{2}+b^{2}}$, where $a = 7$ and $b = 24$. So, $c=\sqrt{7^{2}+24^{2}}=\sqrt{49 + 576}=\sqrt{625}=25$.

Step2: Calculate $\tan(D)$

For $\angle D$, the opposite side is $EF = 24$ and the adjacent side is $DF=7$. So, $\tan(D)=\frac{EF}{DF}=\frac{24}{7}$.

Step3: Calculate $\sin(E)$

For $\angle E$, the opposite side is $DF = 7$ and the hypotenuse is $DE = 25$. So, $\sin(E)=\frac{DF}{DE}=\frac{7}{25}$.

Step4: Calculate $\tan(D)$ again (re - emphasizing)

$\tan(D)=\frac{EF}{DF}=\frac{24}{7}$ as before.

Step5: Calculate $\cos(E)$

For $\angle E$, the adjacent side is $EF = 24$ and the hypotenuse is $DE = 25$. So, $\cos(E)=\frac{EF}{DE}=\frac{24}{25}$.

Step6: Calculate $\sin(D)$

For $\angle D$, the opposite side is $EF = 24$ and the hypotenuse is $DE = 25$. So, $\sin(D)=\frac{EF}{DE}=\frac{24}{25}$.

Answer:

$\tan(D)=\frac{24}{7}$, $\sin(E)=\frac{7}{25}$, $\cos(E)=\frac{24}{25}$