Sketch an angle θ in standard position such that θ has the least positive measure, and the given point is on the terminal side of θ. Then find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. See Examples 1, 2, and 4. (―12 , ―5)

Verified step by step guidance

1. Step 1: Understand the problem. We are given a point (-12, -5) on the terminal side of an angle \( \theta \) in standard position. The goal is to sketch this angle and find the six trigonometric functions: sine, cosine, tangent, cosecant, secant, and cotangent.
2. Step 2: Sketch the angle. Since the point (-12, -5) lies in the third quadrant (both x and y are negative), the terminal side of \( \theta \) is in the third quadrant. The angle \( \theta \) is measured from the positive x-axis to this terminal side, moving counterclockwise.
3. Step 3: Calculate the radius (or hypotenuse) \( r \) using the distance formula: \[ r = \sqrt{x^2 + y^2} = \sqrt{(-12)^2 + (-5)^2} = \sqrt{144 + 25} = \sqrt{169} \].
4. Step 4: Find the six trigonometric functions using the definitions based on the coordinates and radius: - \( \sin \theta = \frac{y}{r} = \frac{-5}{r} \) - \( \cos \theta = \frac{x}{r} = \frac{-12}{r} \) - \( \tan \theta = \frac{y}{x} = \frac{-5}{-12} \) - \( \csc \theta = \frac{r}{y} = \frac{r}{-5} \) - \( \sec \theta = \frac{r}{x} = \frac{r}{-12} \) - \( \cot \theta = \frac{x}{y} = \frac{-12}{-5} \)
5. Step 5: Rationalize denominators if necessary and simplify each trigonometric function to its simplest form, keeping in mind the signs based on the quadrant where the angle lies.

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 determined by rotating the initial side counterclockwise by the angle measure θ. Sketching the angle involves plotting the given point on the terminal side and measuring the angle from the positive x-axis to that point.

Finding the Reference Angle and Least Positive Angle

The least positive angle θ is the smallest positive rotation from the positive x-axis to the terminal side containing the point. To find θ, calculate the reference angle using the coordinates of the point and then determine the quadrant to adjust θ accordingly. This ensures the angle measure is between 0° and 360° (or 0 and 2π radians).

Six Trigonometric Functions from Coordinates

Given a point (x, y) on the terminal side, the six trigonometric functions—sine, cosine, tangent, cosecant, secant, and cotangent—can be found using x, y, and the radius r = √(x² + y²). For example, sin θ = y/r and cos θ = x/r. Rationalizing denominators ensures the answers are in simplified, standard form.