Question

2.) find the exact value of the following trigonometric functions. be sure to rationalize denominators where necessary. if the value is undefined, so state.

\\( \sin ( 120 ^ { \circ } ) = \\)____
\\( \cos ( - 225 ^ { \circ } ) = \\)____
\\( \tan ( 210 ^ { \circ } ) = \\)____
\\( \sec ( 330 ^ { \circ } ) = \\)____
\\( \csc ( 180 ^ { \circ } ) = \\)____
\\( \cot ( - 315 ^ { \circ } ) = \\)____

3.) find the exact value of the following trigonometric functions. be sure to rationalize denominators where necessary. if the value is undefined, so state.

\\( \sin \left( \frac { 5 \pi } { 3 } \
ight) = \\)____
\\( \cos \left( - \frac { 3 \pi } { 2 } \
ight) = \\)____
\\( \tan \left( - \frac { 5 \pi } { 6 } \
ight) = \\)____
\\( \sec ( 2 \pi ) = \\)____
\\( \csc \left( \frac { 5 \pi } { 4 } \
ight) = \\)____
\\( \cot \left( \frac { 2 \pi } { 3 } \
ight) = \\)____

Explanation:

Step1: Find \(\sin(120^{\circ})\)

\(120^{\circ}=180^{\circ} - 60^{\circ}\), \(\sin(120^{\circ})=\sin(180^{\circ}-60^{\circ})\)
Using the identity \(\sin(A - B)=\sin A\cos B-\cos A\sin B\) (here \(A = 180^{\circ}\), \(B=60^{\circ}\)), \(\sin(180^{\circ}-60^{\circ})=\sin60^{\circ}\)
\(\sin60^{\circ}=\frac{\sqrt{3}}{2}\)

Step2: Find \(\cos(-225^{\circ})\)

Using the identity \(\cos(-\theta)=\cos\theta\), so \(\cos(-225^{\circ})=\cos225^{\circ}\)
\(225^{\circ}=180^{\circ}+45^{\circ}\), \(\cos(180^{\circ}+\theta)=-\cos\theta\) (here \(\theta = 45^{\circ}\))
\(\cos225^{\circ}=-\cos45^{\circ}=-\frac{\sqrt{2}}{2}\)

Step3: Find \(\tan(210^{\circ})\)

\(210^{\circ}=180^{\circ}+30^{\circ}\), \(\tan(180^{\circ}+\theta)=\tan\theta\) (here \(\theta = 30^{\circ}\))
\(\tan210^{\circ}=\tan30^{\circ}=\frac{\sqrt{3}}{3}\)

Step4: Find \(\sec(330^{\circ})\)

\(330^{\circ}=360^{\circ}-30^{\circ}\), \(\sec\theta=\frac{1}{\cos\theta}\), \(\cos(360^{\circ}-\theta)=\cos\theta\) (here \(\theta = 30^{\circ}\))
\(\cos330^{\circ}=\cos30^{\circ}=\frac{\sqrt{3}}{2}\), so \(\sec330^{\circ}=\frac{2}{\sqrt{3}}=\frac{2\sqrt{3}}{3}\)

Step5: Find \(\csc(180^{\circ})\)

\(\csc\theta=\frac{1}{\sin\theta}\), \(\sin180^{\circ}=0\), so \(\csc(180^{\circ})\) is undefined

Step6: Find \(\cot(-315^{\circ})\)

Using the identity \(\cot(-\theta)=-\cot\theta\), so \(\cot(-315^{\circ})=-\cot315^{\circ}\)
\(315^{\circ}=360^{\circ}-45^{\circ}\), \(\cot(360^{\circ}-\theta)=-\cot\theta\) (here \(\theta = 45^{\circ}\))
\(\cot315^{\circ}=-\cot45^{\circ}=- 1\), so \(\cot(-315^{\circ})=-(-1) = 1\)

Step7: Find \(\sin(\frac{5\pi}{3})\)

\(\frac{5\pi}{3}=2\pi-\frac{\pi}{3}\), \(\sin(2\pi-\theta)=-\sin\theta\) (here \(\theta=\frac{\pi}{3}\))
\(\sin\frac{\pi}{3}=\frac{\sqrt{3}}{2}\), so \(\sin\frac{5\pi}{3}=-\frac{\sqrt{3}}{2}\)

Step8: Find \(\cos(-\frac{3\pi}{2})\)

Using the identity \(\cos(-\theta)=\cos\theta\), so \(\cos(-\frac{3\pi}{2})=\cos\frac{3\pi}{2}=0\)

Step9: Find \(\tan(-\frac{5\pi}{6})\)

Using the identity \(\tan(-\theta)=-\tan\theta\), so \(\tan(-\frac{5\pi}{6})=-\tan\frac{5\pi}{6}\)
\(\frac{5\pi}{6}=\pi-\frac{\pi}{6}\), \(\tan(\pi-\theta)=-\tan\theta\) (here \(\theta=\frac{\pi}{6}\))
\(\tan\frac{5\pi}{6}=-\tan\frac{\pi}{6}=-\frac{\sqrt{3}}{3}\), so \(\tan(-\frac{5\pi}{6})=-(-\frac{\sqrt{3}}{3})=\frac{\sqrt{3}}{3}\)

Step10: Find \(\sec(2\pi)\)

\(\sec\theta=\frac{1}{\cos\theta}\), \(\cos(2\pi)=1\), so \(\sec(2\pi)=1\)

Step11: Find \(\csc(\frac{5\pi}{4})\)

\(\frac{5\pi}{4}=\pi+\frac{\pi}{4}\), \(\csc\theta=\frac{1}{\sin\theta}\), \(\sin(\pi+\theta)=-\sin\theta\) (here \(\theta=\frac{\pi}{4}\))
\(\sin\frac{5\pi}{4}=-\sin\frac{\pi}{4}=-\frac{\sqrt{2}}{2}\), so \(\csc\frac{5\pi}{4}=-\sqrt{2}\)

Step12: Find \(\cot(\frac{2\pi}{3})\)

\(\frac{2\pi}{3}=\pi-\frac{\pi}{3}\), \(\cot(\pi-\theta)=-\cot\theta\) (here \(\theta=\frac{\pi}{3}\))
\(\cot\frac{\pi}{3}=\frac{\sqrt{3}}{3}\), so \(\cot\frac{2\pi}{3}=-\frac{\sqrt{3}}{3}\)

Answer:

\(\sin(120^{\circ})=\frac{\sqrt{3}}{2}\), \(\cos(-225^{\circ})=-\frac{\sqrt{2}}{2}\), \(\tan(210^{\circ})=\frac{\sqrt{3}}{3}\), \(\sec(330^{\circ})=\frac{2\sqrt{3}}{3}\), \(\csc(180^{\circ})\) is undefined, \(\cot(-315^{\circ}) = 1\), \(\sin(\frac{5\pi}{3})=-\frac{\sqrt{3}}{2}\), \(\cos(-\frac{3\pi}{2}) = 0\), \(\tan(-\frac{5\pi}{6})=\frac{\sqrt{3}}{3}\), \(\sec(2\pi)=1\), \(\csc(\frac{5\pi}{4})=-\sqrt{2}\), \(\cot(\frac{2\pi}{3})=-\frac{\sqrt{3}}{3}\)