CONCEPT PREVIEW The terminal side of an angle θ in standard position passes through the point (― 3,― I3) Use the figure to find the following values. Rationalize denominators when applicable. tan θ

Verified step by step guidance

1. Identify the coordinates of the point through which the terminal side of the angle \( \theta \) passes, which are \((-3, -\sqrt{3})\).
2. Recall that the tangent of an angle \( \theta \) in standard position is given by \( \tan \theta = \frac{y}{x} \), where \( (x, y) \) are the coordinates of the point on the terminal side.
3. Substitute the coordinates \((-3, -\sqrt{3})\) into the formula: \( \tan \theta = \frac{-\sqrt{3}}{-3} \).
4. Simplify the expression \( \frac{-\sqrt{3}}{-3} \) by canceling the negative signs, resulting in \( \frac{\sqrt{3}}{3} \).
5. Rationalize the denominator if necessary by multiplying the numerator and the denominator by \( \sqrt{3} \) to ensure the denominator is a rational number.

Key Concepts

Standard Position of an Angle

An angle is said to be in standard position when its vertex is at the origin of a coordinate system and its initial side lies along the positive x-axis. The terminal side of the angle is formed by rotating the initial side counterclockwise. Understanding this concept is crucial for determining the coordinates of points that the terminal side intersects, which is essential for calculating trigonometric functions.

Coordinates and Trigonometric Ratios

The coordinates of a point on the terminal side of an angle provide the necessary values to compute trigonometric ratios. For a point (x, y), the tangent of the angle θ can be defined as the ratio of the y-coordinate to the x-coordinate, or tan(θ) = y/x. This relationship is fundamental for solving problems involving angles and their corresponding trigonometric functions.

Rationalizing Denominators

Rationalizing the denominator is a mathematical technique used to eliminate any radical expressions from the denominator of a fraction. This is often done by multiplying the numerator and denominator by a suitable value that will simplify the expression. In trigonometry, rationalizing is important for presenting final answers in a standard form, especially when dealing with square roots in trigonometric calculations.