Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. 405°

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Step 1: Recognize that the angle 405° is greater than 360°, so find its reference angle by subtracting 360°: \(405^\circ - 360^\circ = 45^\circ\).

Step 2: Determine the quadrant where 405° lies. Since 405° is 45° past 360°, it lies in the first quadrant, where all trigonometric functions are positive.

Step 3: Recall the exact trigonometric values for 45°: \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), and \(\tan 45^\circ = 1\).

Step 4: Use the reference angle values and the quadrant sign to find the six trigonometric functions for 405°: \(\sin 405^\circ\), \(\cos 405^\circ\), \(\tan 405^\circ\), \(\csc 405^\circ\), \(\sec 405^\circ\), and \(\cot 405^\circ\).

Step 5: Rationalize denominators where necessary, for example, rewrite \(\frac{1}{\frac{\sqrt{2}}{2}}\) as \(\frac{\sqrt{2}}{1}\) to express the reciprocal functions in simplest form.

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