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

Verified step by step guidance

1. Step 1: Recognize that the angle 405° is greater than 360°, so find its reference angle by subtracting 360°: \(405^\circ - 360^\circ = 45^\circ\).
2. 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.
3. Step 3: Recall the exact trigonometric values for 45°: \(\sin 45^\circ = \frac{\sqrt{2}}{2}\), \(\cos 45^\circ = \frac{\sqrt{2}}{2}\), and \(\tan 45^\circ = 1\).
4. 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 \(\cot 405^\circ\).
5. 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.

Key Concepts

Reference Angles and Angle Reduction

Angles greater than 360° can be reduced by subtracting 360° to find a coterminal angle within the standard 0° to 360° range. This helps identify the reference angle, which is the acute angle the terminal side makes with the x-axis, essential for determining trigonometric values.

Signs of Trigonometric Functions in Quadrants

The sign of sine, cosine, and tangent depends on the quadrant where the angle's terminal side lies. Knowing the quadrant after angle reduction allows correct assignment of positive or negative values to each trigonometric function.

Exact Values and Rationalizing Denominators

Exact trigonometric values often involve square roots and fractions. Rationalizing denominators means rewriting expressions to eliminate radicals from the denominator, providing a simplified and standardized form of the answer.