Question

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

Explanation:

Step1: Find the length of \( GF \)

Use the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \). Here \( c = 53 \), \( b = 45 \), and \( a=GF \).
\( GF=\sqrt{53^{2}-45^{2}}=\sqrt{(53 + 45)(53 - 45)}=\sqrt{98\times8}=\sqrt{784}=28 \)

Step2: Calculate \( \sin(G) \)

\(\sin\theta=\frac{\text{opposite}}{\text{hypotenuse}}\). For \( \angle G \), the opposite side to \( \angle G \) is \( FE = 45 \), and the hypotenuse \( GE = 53 \).
\(\sin(G)=\frac{45}{53}\)

Step3: Calculate \( \cos(G) \)

\(\cos\theta=\frac{\text{adjacent}}{\text{hypotenuse}}\). For \( \angle G \), the adjacent side to \( \angle G \) is \( GF = 28 \), and the hypotenuse \( GE = 53 \).
\(\cos(G)=\frac{28}{53}\)

Step4: Calculate \( \tan(G) \)

\(\tan\theta=\frac{\text{opposite}}{\text{adjacent}}\). For \( \angle G \), the opposite side to \( \angle G \) is \( FE = 45 \), and the adjacent side to \( \angle G \) is \( GF = 28 \).
\(\tan(G)=\frac{45}{28}\)

Answer:

\(\sin(G)=\frac{45}{53}\), \(\cos(G)=\frac{28}{53}\), \(\tan(G)=\frac{45}{28}\)