Find exact values of the six trigonometric functions of each angle. Rationalize denominators when applicable. See Examples 2, 3, and 5. -1860°

Verified step by step guidance

1. Step 1: Understand that trigonometric functions are periodic, meaning their values repeat after a certain interval. For sine, cosine, and tangent, the period is 360°. To find the equivalent angle between 0° and 360°, reduce the given angle by adding or subtracting multiples of 360° until the angle lies within this range.
2. Step 2: Calculate the equivalent angle for -1860° by adding 360° repeatedly: \(-1860° + 360° \times n = \theta\), where \(\theta\) is between 0° and 360°. Find the integer \(n\) that satisfies this condition.
3. Step 3: Once you have the equivalent angle \(\theta\), use the unit circle or known trigonometric values to find the sine, cosine, and tangent of \(\theta\). Remember that the six trigonometric functions are sine, cosine, tangent, cosecant, secant, and cotangent.
4. Step 4: Use the definitions of the reciprocal functions to find cosecant, secant, and cotangent: \(\csc \theta = \frac{1}{\sin \theta}\), \(\sec \theta = \frac{1}{\cos \theta}\), and \(\cot \theta = \frac{1}{\tan \theta}\). Make sure to rationalize denominators if necessary.
5. Step 5: Write down the exact values of all six trigonometric functions for the angle \(\theta\), ensuring that any radicals in denominators are rationalized to complete the solution.

Key Concepts

Angle Coterminality and Reduction

Angles differing by full rotations (360°) share the same terminal side and thus the same trigonometric values. To find the trigonometric functions of -1860°, reduce the angle by adding or subtracting multiples of 360° until it lies within the standard 0° to 360° range.

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 coordinates. Understanding their definitions and relationships is essential to compute exact values once the angle is simplified.

Rationalizing Denominators

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