Question

find the secant of ∠f.

simplify your answer and write it as a proper fraction, improper fraction, or whole number.
sec(f) =
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Explanation:

Step1: Recall the definition of secant

The secant of an angle in a right triangle is the reciprocal of the cosine of that angle. For an angle \( \theta \), \( \sec(\theta) = \frac{1}{\cos(\theta)} \), and \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \), so \( \sec(\theta) = \frac{\text{hypotenuse}}{\text{adjacent}} \).

Step2: Identify the sides relative to \( \angle F \)

In right triangle \( EFG \) (with right angle at \( E \)):

• The hypotenuse is \( GF \). First, we need to find the length of \( GF \) using the Pythagorean theorem. The legs are \( EG = 4 \) and \( EF = 3 \). So, by Pythagoras: \( GF^2 = EG^2 + EF^2 = 4^2 + 3^2 = 16 + 9 = 25 \), so \( GF = \sqrt{25} = 5 \).
• The adjacent side to \( \angle F \) is \( EF = 3 \).
• The hypotenuse is \( GF = 5 \).

Step3: Calculate \( \sec(F) \)

Using the definition \( \sec(F) = \frac{\text{hypotenuse}}{\text{adjacent}} \), we substitute the values: \( \sec(F) = \frac{5}{3} \).

Answer:

\( \frac{5}{3} \)