Question

find the sine, cosine, and tangent of ∠b.

simplify your answers and write them as proper fractions, improper fractions, or whole numbers.

sin(b) =
cos(b) =
tan(b) =

Explanation:

Step1: Find the hypotenuse \( BD \)

In right triangle \( BCD \), by Pythagorean theorem \( BD^2 = BC^2 + CD^2 \). Given \( BC = 48 \), \( CD = 55 \), so \( BD = \sqrt{48^2 + 55^2} = \sqrt{2304 + 3025} = \sqrt{5329} = 73 \).

Step2: Calculate \( \sin(B) \)

\( \sin(B) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{CD}{BD} = \frac{55}{73} \).

Step3: Calculate \( \cos(B) \)

\( \cos(B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{BD} = \frac{48}{73} \).

Step4: Calculate \( \tan(B) \)

\( \tan(B) = \frac{\text{opposite}}{\text{adjacent}} = \frac{CD}{BC} = \frac{55}{48} \).

Answer:

\( \sin(B) = \frac{55}{73} \)
\( \cos(B) = \frac{48}{73} \)
\( \tan(B) = \frac{55}{48} \)