In Exercises 61–86, use reference angles to find the exact value of each expression. Do not use a calculator. tan(-𝜋/4)

Verified step by step guidance

1. Recall that the tangent function is periodic and odd, meaning that \(\tan(-\theta) = -\tan(\theta)\). This property will help simplify \(\tan(-\pi/4)\).
2. Identify the reference angle for \(-\pi/4\). Since the angle is negative, its reference angle is the positive acute angle \(\pi/4\).
3. Find the value of \(\tan(\pi/4)\). From the unit circle or special triangles, \(\tan(\pi/4) = 1\).
4. Apply the odd function property: \(\tan(-\pi/4) = -\tan(\pi/4) = -1\).
5. Thus, the exact value of \(\tan(-\pi/4)\) is \(-1\).

Key Concepts

Reference Angles

A reference angle is the acute angle formed between the terminal side of an angle and the x-axis. It helps simplify trigonometric calculations by relating any angle to a corresponding angle in the first quadrant, where trigonometric values are well-known.

Tangent Function and Its Properties

The tangent of an angle in the unit circle is the ratio of the sine to the cosine of that angle. It is periodic with period π and odd, meaning tan(-θ) = -tan(θ), which is useful for evaluating tangent of negative angles without a calculator.

Exact Values of Special Angles

Certain angles like π/4, π/3, and π/6 have known exact trigonometric values. For π/4, tan(π/4) = 1, so using these exact values avoids approximation and calculator use, enabling precise answers for trigonometric expressions.