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. Here, the point is \( (-3, -\sqrt{3}) \).
2. Recall that \( \tan \theta = \frac{y}{x} \), where \( x \) and \( y \) are the coordinates of the point on the terminal side of the angle.
3. Substitute the values of \( x = -3 \) and \( y = -\sqrt{3} \) into the tangent formula: \( \tan \theta = \frac{-\sqrt{3}}{-3} \).
4. Simplify the fraction by dividing numerator and denominator, and then rationalize the denominator if necessary.
5. Express the final simplified form of \( \tan \theta \) with a rationalized denominator.

Key Concepts

Standard Position of an Angle

An angle is in standard position when its vertex is at the origin and its initial side lies along the positive x-axis. The terminal side is the ray that rotates from the initial side to form the angle θ. Understanding this helps locate the angle based on a point through which its terminal side passes.

Coordinates and Trigonometric Ratios

The coordinates (x, y) of a point on the terminal side of an angle can be used to find trigonometric ratios. Specifically, tan θ is the ratio of y to x (tan θ = y/x). Knowing the point allows direct calculation of tangent and other trigonometric functions.

Rationalizing Denominators

Rationalizing the denominator involves eliminating any square roots or irrational numbers from the denominator of a fraction. This is done by multiplying numerator and denominator by a suitable expression, making the expression simpler and more standardized in trigonometry answers.