Question

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

Explanation:

Step1: Find the length of VW

In right triangle \( UVW \) with right angle at \( V \), by the Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( c = 85 \) (hypotenuse), \( a = 77 \) (one leg), and \( b = VW \) (the other leg). So we have \( 77^2 + VW^2 = 85^2 \). Calculating \( 77^2 = 5929 \) and \( 85^2 = 7225 \). Then \( VW^2 = 7225 - 5929 = 1296 \), so \( VW = \sqrt{1296} = 36 \).

Step2: Calculate \( \sin(W) \)

For angle \( W \), the sine is the ratio of the opposite side to the hypotenuse. The opposite side to \( \angle W \) is \( UV = 77 \), and the hypotenuse is \( UW = 85 \). So \( \sin(W) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{77}{85} \).

Step3: Calculate \( \cos(W) \)

The cosine of angle \( W \) is the ratio of the adjacent side to the hypotenuse. The adjacent side to \( \angle W \) is \( VW = 36 \), and the hypotenuse is \( UW = 85 \). So \( \cos(W) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{36}{85} \).

Step4: Calculate \( \tan(W) \)

The tangent of angle \( W \) is the ratio of the opposite side to the adjacent side. So \( \tan(W) = \frac{\text{opposite}}{\text{adjacent}} = \frac{77}{36} \).

Answer:

\( \sin(W) = \frac{77}{85} \)
\( \cos(W) = \frac{36}{85} \)
\( \tan(W) = \frac{77}{36} \)