Question

find the sine, cosine, and tangent of \\( \angle v \\).
\\( v \\)---33---\\( w \\) (right angle at \\( w \\)), \\( vu = 65 \\), \\( u \\) is the third vertex. 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 \( V W U \), by the Pythagorean theorem \( a^2 + b^2 = c^2 \), where \( c = 65 \) (hypotenuse), \( a = 33 \) (one leg), and \( b = UW \) (the other leg). So \( UW=\sqrt{65^{2}-33^{2}}=\sqrt{4225 - 1089}=\sqrt{3136} = 56 \).

Step2: Calculate \( \sin(V) \)

The sine of an angle in a right triangle is \( \frac{\text{opposite}}{\text{hypotenuse}} \). For \( \angle V \), the opposite side is \( UW = 56 \), and the hypotenuse is \( VU = 65 \). So \( \sin(V)=\frac{56}{65} \).

Step3: Calculate \( \cos(V) \)

The cosine of an angle in a right triangle is \( \frac{\text{adjacent}}{\text{hypotenuse}} \). For \( \angle V \), the adjacent side is \( VW = 33 \), and the hypotenuse is \( VU = 65 \). So \( \cos(V)=\frac{33}{65} \).

Step4: Calculate \( \tan(V) \)

The tangent of an angle in a right triangle is \( \frac{\text{opposite}}{\text{adjacent}} \). For \( \angle V \), the opposite side is \( UW = 56 \), and the adjacent side is \( VW = 33 \). So \( \tan(V)=\frac{56}{33} \).

Answer:

\( \sin(V)=\frac{56}{65} \)
\( \cos(V)=\frac{33}{65} \)
\( \tan(V)=\frac{56}{33} \)