Question

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

Explanation:

Step1: Find the length of VW

In right triangle \( V W X \), by the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( c = 97 \) (hypotenuse), \( b = 65 \) (one leg), and \( a=VW \) (the other leg). So \( VW=\sqrt{97^{2}-65^{2}}=\sqrt{(97 + 65)(97 - 65)}=\sqrt{162\times32}=\sqrt{5184}=72 \).

Step2: Find \( \sin(X) \)

For angle \( X \), the sine of an angle in a right triangle is \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \). The side opposite \( \angle X \) is \( VW = 72 \), and the hypotenuse is \( VX=97 \). So \( \sin(X)=\frac{72}{97} \).

Step3: Find \( \cos(X) \)

The cosine of an angle in a right triangle is \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \). The side adjacent to \( \angle X \) is \( XW = 65 \), and the hypotenuse is \( VX = 97 \). So \( \cos(X)=\frac{65}{97} \).

Step4: Find \( \tan(X) \)

The tangent of an angle in a right triangle is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \). The side opposite \( \angle X \) is \( VW = 72 \), and the side adjacent to \( \angle X \) is \( XW = 65 \). So \( \tan(X)=\frac{72}{65} \).

Answer:

\( \sin(X)=\frac{72}{97} \)
\( \cos(X)=\frac{65}{97} \)
\( \tan(X)=\frac{72}{65} \)