Question

what is sec(∠b)? reduce fractional answers to lowest terms.

Explanation:

Step1: Recall secant definition

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 \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \) in a right triangle.

Step2: Identify sides for \( \angle B \)

In right triangle \( ABC \) with right angle at \( C \), for \( \angle B \):

• The adjacent side to \( \angle B \) is \( BC = 3 \).
• The hypotenuse is \( AB = 5 \).

Step3: Calculate \( \cos(\angle B) \)

Using the cosine definition: \( \cos(\angle B) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{BC}{AB} = \frac{3}{5} \).

Step4: Calculate \( \sec(\angle B) \)

Since \( \sec(\angle B) = \frac{1}{\cos(\angle B)} \), substitute \( \cos(\angle B) = \frac{3}{5} \):
\( \sec(\angle B) = \frac{1}{\frac{3}{5}} = \frac{5}{3} \).

Answer:

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