Question

find the sine, cosine, and tangent of $\angle v$. simplify your answers and write them as proper fractions, improper fractions, or whole numbers. $\sin(v) = \square$ $\cos(v) = \square$ $\tan(v) = \square$

Explanation:

Step1: Find the length of UW

In right triangle \( UVW \) with \( \angle U = 90^\circ \), \( VU = 5 \), \( VW = 13 \). By Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( c = VW = 13 \), \( b = VU = 5 \), and \( a = UW \). So \( UW = \sqrt{VW^2 - VU^2} = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \).

Step2: Find \( \sin(V) \)

Sine of an angle in a right triangle is \( \frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle V \), opposite side is \( UW = 12 \), hypotenuse is \( VW = 13 \). So \( \sin(V) = \frac{UW}{VW} = \frac{12}{13} \).

Step3: Find \( \cos(V) \)

Cosine of an angle in a right triangle is \( \frac{\text{adjacent}}{\text{hypotenuse}} \). For \( \angle V \), adjacent side is \( VU = 5 \), hypotenuse is \( VW = 13 \). So \( \cos(V) = \frac{VU}{VW} = \frac{5}{13} \).

Step4: Find \( \tan(V) \)

Tangent of an angle in a right triangle is \( \frac{\text{opposite}}{\text{adjacent}} \). For \( \angle V \), opposite side is \( UW = 12 \), adjacent side is \( VU = 5 \). So \( \tan(V) = \frac{UW}{VU} = \frac{12}{5} \).

Answer:

\( \sin(V) = \frac{12}{13} \)
\( \cos(V) = \frac{5}{13} \)
\( \tan(V) = \frac{12}{5} \)