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

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

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

Explanation:

Identify reciprocal relationships

Using the Reciprocal Trigonometric Functions knowledge point, we find the remaining reciprocal values:

• \(\sec(\theta) = \frac{1}{\cos(\theta)}\)
• \(\csc(\theta) = \frac{1}{\sin(\theta)}\)
• \(\cot(\theta) = \frac{1}{\tan(\theta)}\)

Calculate values for 45 degrees

Using the Trigonometric Ratios knowledge point:

• \(\sec(45^\circ) = \frac{1}{\cos(45^\circ)} = \frac{2}{\sqrt{2}} = \sqrt{2}\)
• \(\csc(45^\circ) = \frac{1}{\sin(45^\circ)} = \frac{2}{\sqrt{2}} = \sqrt{2}\)
• \(\cot(45^\circ) = \frac{1}{\tan(45^\circ)} = \frac{1}{1} = 1\)

Calculate values for 60 degrees

Using the Trigonometric Ratios knowledge point:

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

Answer:

Angle \(\theta\)	Function	Unsimplified Value	Simplified/Rationalized Value
\(45^\circ\)	\(\csc(45^\circ)\)	\(\frac{2}{\sqrt{2}}\)	\(\sqrt{2}\)
\(45^\circ\)	\(\cot(45^\circ)\)	\(\frac{1}{1}\)	\(1\)
\(60^\circ\)	\(\sec(60^\circ)\)	\(\frac{1}{1/2}\)	\(2\)
\(60^\circ\)	\(\csc(60^\circ)\)	\(\frac{2}{\sqrt{3}}\)	\(\frac{2\sqrt{3}}{3}\)
\(60^\circ\)	\(\cot(60^\circ)\)	\(\frac{1}{\sqrt{3}}\)	\(\frac{\sqrt{3}}{3}\)