Question

find the sine, cosine, and tangent of ∠d. simplify your answers and write them as proper fractions, improper fractions, or whole numbers. sin(d) = cos(d) = tan(d) =

Explanation:

Step1: Find the length of CE

In right triangle \( CDE \), we know the hypotenuse \( CD = 53 \) and the leg \( DE = 28 \). Using the Pythagorean theorem \( CE=\sqrt{CD^{2}-DE^{2}} \).

$$ CE=\sqrt{53^{2}-28^{2}}=\sqrt{(53 + 28)(53 - 28)}=\sqrt{81\times25}=\sqrt{2025}=45 $$

Step2: Find \( \sin(D) \)

For angle \( D \), the opposite side is \( CE = 45 \) and the hypotenuse is \( CD = 53 \). So \( \sin(D)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{CE}{CD}=\frac{45}{53} \).

Step3: Find \( \cos(D) \)

For angle \( D \), the adjacent side is \( DE = 28 \) and the hypotenuse is \( CD = 53 \). So \( \cos(D)=\frac{\text{adjacent}}{\text{hypotenuse}}=\frac{DE}{CD}=\frac{28}{53} \).

Step4: Find \( \tan(D) \)

For angle \( D \), the opposite side is \( CE = 45 \) and the adjacent side is \( DE = 28 \). So \( \tan(D)=\frac{\text{opposite}}{\text{adjacent}}=\frac{CE}{DE}=\frac{45}{28} \).

Answer:

\( \sin(D)=\frac{45}{53} \), \( \cos(D)=\frac{28}{53} \), \( \tan(D)=\frac{45}{28} \)