Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -2205°

Verified step by step guidance

1. Step 1: Understand that the angle given is -2205°, which is a negative angle. To find the trigonometric functions, first convert this angle to a positive coterminal angle between 0° and 360° by adding multiples of 360° until the angle lies within this range. Use the formula: \(\theta_{coterminal} = \theta + 360k\), where \(k\) is an integer chosen to make \(\theta_{coterminal}\) between 0° and 360°.
2. Step 2: Calculate the coterminal angle by adding 360° repeatedly to -2205° until the result is between 0° and 360°. This will give you an equivalent angle that has the same trigonometric values as -2205°.
3. Step 3: Once you have the coterminal angle, determine the reference angle. The reference angle is the acute angle formed between the terminal side of the coterminal angle and the x-axis. This helps in finding the exact values of the trigonometric functions.
4. Step 4: Identify the quadrant in which the coterminal angle lies. The signs of the six trigonometric functions (sine, cosine, tangent, cosecant, secant, cotangent) depend on the quadrant. Recall the ASTC rule (All Students Take Calculus) to determine the signs.
5. Step 5: Use known exact values of trigonometric functions for standard angles (like 30°, 45°, 60°, etc.) and apply the appropriate sign based on the quadrant to find the exact values of all six trigonometric functions. Rationalize denominators if necessary.

Key Concepts

Angle Reduction Using Coterminal Angles

Angles larger than 360° or negative angles can be simplified by adding or subtracting multiples of 360° to find a coterminal angle between 0° and 360°. This helps in evaluating trigonometric functions by referencing standard angle values.

Definition of the Six Trigonometric Functions

The six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—are ratios derived from a right triangle or the unit circle. Understanding their definitions and relationships is essential for calculating exact values.

Rationalizing Denominators

Rationalizing denominators involves eliminating radicals from the denominator of a fraction by multiplying numerator and denominator by a suitable expression. This is important for expressing trigonometric values in a simplified, exact form.