Match each expression in Column I with its value in Column II.
8. tan (-π/8)

Verified step by step guidance

1. Recall the definition of the tangent function: \(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\). This will help understand the behavior of \(\tan(-\frac{\pi}{8})\).
2. Use the odd function property of tangent: \(\tan(-\theta) = -\tan(\theta)\). This means \(\tan(-\frac{\pi}{8}) = -\tan(\frac{\pi}{8})\).
3. Focus on finding \(\tan(\frac{\pi}{8})\). Recognize that \(\frac{\pi}{8}\) is \(22.5^\circ\), which is half of \(45^\circ\).
4. Apply the half-angle formula for tangent: \(\tan\left(\frac{\alpha}{2}\right) = \frac{1 - \cos(\alpha)}{\sin(\alpha)}\) or alternatively \(\tan\left(\frac{\alpha}{2}\right) = \sqrt{\frac{1 - \cos(\alpha)}{1 + \cos(\alpha)}}\) depending on the quadrant.
5. Calculate \(\tan(\frac{\pi}{8})\) using the half-angle formula with \(\alpha = \frac{\pi}{4}\), then use the odd function property to find \(\tan(-\frac{\pi}{8})\) by negating the result.

Key Concepts

Tangent Function and Its Properties

The tangent function, tan(θ), is defined as the ratio of the sine and cosine of an angle θ. It is periodic with period π and is odd, meaning tan(-θ) = -tan(θ). Understanding these properties helps evaluate tangent values for negative angles.

Reference Angles and Angle Reduction

Reference angles are acute angles used to find the trigonometric values of angles outside the first quadrant. For negative angles like -π/8, the value of tan(-π/8) can be found by using the odd property of tangent and the positive reference angle π/8.

Exact Values of Special Angles

Certain angles, such as π/8 (22.5°), have known exact trigonometric values expressed in surds or radicals. Knowing or deriving the exact value of tan(π/8) allows precise calculation of tan(-π/8) without a calculator.