Question

what is csc(∠b)? reduce fractional answers to lowest terms. triangle with right angle at c, ac=6, bc=8, vertices a, c, b

Explanation:

Step1: Find hypotenuse AB

In right triangle \( \triangle ABC \) with \( \angle C = 90^\circ \), \( AC = 6 \), \( BC = 8 \). By Pythagorean theorem:
\( AB=\sqrt{AC^{2}+BC^{2}}=\sqrt{6^{2}+8^{2}}=\sqrt{36 + 64}=\sqrt{100}=10 \).

Step2: Recall cosecant definition

\( \csc(\angle B)=\frac{1}{\sin(\angle B)} \), and \( \sin(\angle B)=\frac{\text{opposite}}{\text{hypotenuse}}=\frac{AC}{AB} \).

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

Substitute \( AC = 6 \), \( AB = 10 \):
\( \sin(\angle B)=\frac{6}{10}=\frac{3}{5} \), so \( \csc(\angle B)=\frac{1}{\frac{3}{5}}=\frac{5}{3} \).

Answer:

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