Question

find the sine, cosine, and tangent of ∠f. simplify your answers and write them as proper frac numbers. sin(f) = cos(f) = tan(f) =

Explanation:

Step1: Find the length of side HG using Pythagorean theorem

Let \(HG = a\), \(GF=65\), \(HF = 97\). By \(a=\sqrt{HF^{2}-GF^{2}}\), we have \(a=\sqrt{97^{2}-65^{2}}=\sqrt{(97 + 65)(97 - 65)}=\sqrt{162\times32}=\sqrt{5184}=72\).

Step2: Recall trig - ratio definitions

Sine of an angle in a right - triangle is opposite over hypotenuse, cosine is adjacent over hypotenuse, and tangent is opposite over adjacent. For \(\angle F\), the opposite side is \(HG = 72\), the adjacent side is \(GF = 65\), and the hypotenuse is \(HF=97\).

Step3: Calculate \(\sin(F)\)

\(\sin(F)=\frac{HG}{HF}=\frac{72}{97}\)

Step4: Calculate \(\cos(F)\)

\(\cos(F)=\frac{GF}{HF}=\frac{65}{97}\)

Step5: Calculate \(\tan(F)\)

\(\tan(F)=\frac{HG}{GF}=\frac{72}{65}\)

Answer:

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