Question

written response:

1. determine the following exact values. (1 mark each)

a) \\( \sec 225^{\circ} \\) b) \\( \tan \frac{11 \pi}{6} \\) c) \\( \sin \frac{-7 \pi}{3} \\)
solution: ______ solution: ____ solution: ______

Explanation:

Step1: Find reference angle for \(225^{\circ}\)

\(225^{\circ}- 180^{\circ}=45^{\circ}\). \(\sec\theta=\frac{1}{\cos\theta}\). \(\cos225^{\circ}=-\cos45^{\circ}=-\frac{\sqrt{2}}{2}\). So \(\sec225^{\circ}=\frac{1}{\cos225^{\circ}}=-\sqrt{2}\)

Step2: Find reference angle for \(\frac{11\pi}{6}\)

\(2\pi-\frac{11\pi}{6}=\frac{\pi}{6}\). \(\tan\frac{11\pi}{6}=-\tan\frac{\pi}{6}\). Since \(\tan\frac{\pi}{6}=\frac{\sqrt{3}}{3}\), \(\tan\frac{11\pi}{6}=-\frac{\sqrt{3}}{3}\)

Step3: Find reference angle for \(-\frac{7\pi}{3}\)

\(-\frac{7\pi}{3}+ 2\pi=-\frac{\pi}{3}\), \(\sin(-\frac{7\pi}{3})=\sin(-\frac{\pi}{3})\). Since \(\sin(-\alpha)=-\sin\alpha\), \(\sin(-\frac{\pi}{3})=-\frac{\sqrt{3}}{2}\)

Answer:

a) \(-\sqrt{2}\)
b) \(-\frac{\sqrt{3}}{3}\)
c) \(-\frac{\sqrt{3}}{2}\)