Question

find the secant of \\( \angle w \\).

write your answer in simplified, rationalized form. do not round.

\\( \sec ( w ) = \\)

Explanation:

Step1: Recall the definition of secant

In a right - triangle, \(\sec\theta=\frac{\text{Hypotenuse}}{\text{Adjacent}}\). For \(\angle W\), we first need to find the length of the hypotenuse \(VW\) using the Pythagorean theorem \(a^{2}+b^{2}=c^{2}\), where \(a = 7\sqrt{3}\), \(b=21\). Then \(VW=\sqrt{(7\sqrt{3})^{2}+21^{2}}=\sqrt{147 + 441}=\sqrt{588}=14\sqrt{3}\).

Step2: Calculate \(\sec(W)\)

The adjacent side to \(\angle W\) is \(21\), and the hypotenuse is \(14\sqrt{3}\). So \(\sec(W)=\frac{VW}{XW}=\frac{14\sqrt{3}}{21}\).

Step3: Simplify the fraction

\(\frac{14\sqrt{3}}{21}=\frac{2\sqrt{3}}{3}\) (divide numerator and denominator by \(7\)).

Answer:

\(\frac{2\sqrt{3}}{3}\)