Question

all six trig functions using the unit circle: quadrant i

\\(\theta = 30^\circ\\)
\\(\cos(30) = \frac{\sqrt{3}}{2}\\) \\(\sec(30) = \frac{2}{\sqrt{3}}\\)
\\(\sin(30) = \frac{1}{2}\\) \\(\csc(30) = \frac{2}{1}\\)
\\(\tan(30) = \frac{\sqrt{3}}{3}\\) \\(\cot(30) = \frac{3}{\sqrt{3}}\\)
radian: \\(\theta = \frac{\pi}{6}\\)

\\(\theta = 45^\circ\\)
\\(\cos(45) = \frac{\sqrt{2}}{2}\\) \\(\sec(45) = \frac{2}{\sqrt{2}}\\)
\\(\sin(45) = \frac{\sqrt{2}}{2}\\) \\(\csc(45) = \frac{2}{\sqrt{2}}\\)
\\(\tan(45) = 1\\) \\(\cot(45) = \frac{1}{1}\\)
radian: \\(\theta = \frac{\pi}{4}\\)

\\(\theta = 60^\circ\\)
\\(\cos(60) = \frac{1}{2}\\) \\(\sec(60) = \frac{1}{1/2}\\)
\\(\sin(60) = \frac{\sqrt{3}}{2}\\) \\(\csc(60) = \frac{2}{\sqrt{3}}\\)
\\(\tan(60) = \sqrt{3}\\) \\(\cot(60) = \frac{1}{\sqrt{3}}\\)
radian: \\(\theta = \frac{\pi}{3}\\)

simplify/rationalize:
\\(\frac{2}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{\sqrt{9}} = \frac{2\sqrt{3}}{3}\\)
\\(\frac{2}{1}\\)
\\(\frac{3}{\sqrt{3}}\\)

simplify/rationalize:
\\(\frac{2}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{2\sqrt{2}}{\sqrt{4}} = \frac{2\sqrt{2}}{2} = \sqrt{2}\\)
\\(\frac{2}{\sqrt{2}}\\)
\\(\frac{1}{1}\\)

simplify/rationalize:
\\(\frac{2}{\sqrt{3}}\\)
\\(\frac{1}{\sqrt{3}}\\)

Explanation:

Rationalize the first column expressions

Using the Rationalizing Denominators knowledge point

$$ \frac{2}{1} = 2 $$

$$ \frac{3}{\sqrt{3}} = \frac{3 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{3\sqrt{3}}{3} = \sqrt{3} $$

Rationalize the second column expressions

Using the Rationalizing Denominators knowledge point

$$ \frac{2}{\sqrt{2}} = \frac{2 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{2\sqrt{2}}{2} = \sqrt{2} $$

$$ \frac{1}{1} = 1 $$

Rationalize the third column expressions

Using the Rationalizing Denominators knowledge point

$$ \frac{2}{\sqrt{3}} = \frac{2 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{2\sqrt{3}}{3} $$

$$ \frac{1}{\sqrt{3}} = \frac{1 \cdot \sqrt{3}}{\sqrt{3} \cdot \sqrt{3}} = \frac{\sqrt{3}}{3} $$

Answer:

Column	Original Expression	Simplified/Rationalized Answer
\(\theta = 30^\circ\)	\(\frac{3}{\sqrt{3}}\)	\(\sqrt{3}\)
\(\theta = 45^\circ\)	\(\frac{2}{\sqrt{2}}\)	\(\sqrt{2}\)
\(\theta = 45^\circ\)	\(\frac{1}{1}\)	\(1\)
\(\theta = 60^\circ\)	\(\frac{2}{\sqrt{3}}\)	\(\frac{2\sqrt{3}}{3}\)
\(\theta = 60^\circ\)	\(\frac{1}{\sqrt{3}}\)	\(\frac{\sqrt{3}}{3}\)