Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Sketch an angle \(\theta\) in standard position such that \(\theta\) has the least possible positive measure, and the given point is on the terminal side of \(\theta .\) Find the values of the six trigonometric functions for each angle. Rationalize denominators when applicable. Do not use a calculator. $$(-2 \sqrt{3}, 2)$$

Step-by-Step Solution

Answer

The angle \(\theta\) is in Quadrant II. \(\sin \theta = \frac{1}{2}\), \(\cos \theta = -\frac{\sqrt{3}}{2}\), \(\tan \theta = -\frac{\sqrt{3}}{3}\), \(\csc \theta = 2\), \(\sec \theta = -\frac{2\sqrt{3}}{3}\), \(\cot \theta = -\sqrt{3}\).

Step 1: Understand the Problem

We are given a point \((-2 \sqrt{3}, 2)\) in the coordinate plane and need to find an angle \(\theta\) in standard position with this point on its terminal side. We also need to find the six trigonometric functions of \(\theta\).

Step 2: Determine the Quadrant

The point \((-2\sqrt{3}, 2)\) has negative x-coordinate and positive y-coordinate, which places it in Quadrant II.

Step 3: Find Radius

Calculate the distance from the origin to the point, which is the radius \(r\). Use the formula: \(r = \sqrt{x^2 + y^2}\). So, \(r = \sqrt{(-2\sqrt{3})^2 + 2^2} = \sqrt{12 + 4} = \sqrt{16} = 4\).

Step 4: Calculate Sine and Cosine

Use the definitions of sine and cosine: \(\sin \theta = \frac{y}{r}\) and \(\cos \theta = \frac{x}{r}\). Thus, \(\sin \theta = \frac{2}{4} = \frac{1}{2}\) and \(\cos \theta = \frac{-2\sqrt{3}}{4} = -\frac{\sqrt{3}}{2}\).

Step 5: Calculate Other Trigonometric Functions

Find other trigonometric functions using sine and cosine. \(\tan \theta = \frac{y}{x} = -\frac{1}{\sqrt{3}}\) and rationalize to get \(-\frac{\sqrt{3}}{3}\). \(\csc \theta = \frac{1}{\sin \theta} = 2\). \(\sec \theta = \frac{1}{\cos \theta} = -\frac{2}{\sqrt{3}}\), rationalize to get \(-\frac{2\sqrt{3}}{3}\). \(\cot \theta = \frac{1}{\tan \theta} = -\sqrt{3}\).

Step 6: Summary of Trigonometric Functions

The six trigonometric function values are: \(\sin \theta = \frac{1}{2}\), \(\cos \theta = -\frac{\sqrt{3}}{2}\), \(\tan \theta = -\frac{\sqrt{3}}{3}\), \(\csc \theta = 2\), \(\sec \theta = -\frac{2\sqrt{3}}{3}\), and \(\cot \theta = -\sqrt{3}\).

Key Concepts

Standard Position

An angle is said to be in standard position when its vertex is located at the origin of a coordinate plane and its initial side is along the positive x-axis. This is a common reference point for measuring angles, making it easier to visualize and compute trigonometric values.
For example, in our problem, the angle \( \theta \) is in standard position, with the point \((-2\sqrt{3}, 2)\) lying on its terminal side. The terminal side of an angle refers to the ray that rotates from the initial side to create the angle.
Using standard position simplifies the process of identifying which quadrant an angle lies in and aids in calculating the trigonometric functions by connecting them to x and y coordinates.

Quadrant

The coordinate plane is divided into four parts called quadrants:

• Quadrant I: positive x and y values
• Quadrant II: negative x and positive y values
• Quadrant III: negative x and y values
• Quadrant IV: positive x and negative y values

In our specific case, the point \((-2\sqrt{3}, 2)\) leads us to Quadrant II.
This quadrant is characterized by x being negative and y being positive. Keeping track of which quadrant a point is in helps determine the signs of trigonometric functions like sine, cosine, and tangent.
For instance, in Quadrant II, sine values are positive, while cosine and tangent values are negative. This helps confirm that \( \sin \theta = \frac{1}{2} \) and \( \cos \theta = -\frac{\sqrt{3}}{2} \) are correct.

Rationalizing Denominators

Rationalizing a denominator refers to the process of eliminating any radical numbers from the denominator of a fraction. To do this, one can multiply both the numerator and the denominator by the same radical.
For example, consider \( \tan \theta = -\frac{1}{\sqrt{3}} \). To rationalize, multiply both the numerator and the denominator by \( \sqrt{3} \) to get \(-\frac{\sqrt{3}}{3} \).
Similarly, for \( \sec \theta = -\frac{2}{\sqrt{3}} \), rationalizing yields \(-\frac{2\sqrt{3}}{3} \).Rationalizing ensures that our expressions do not have roots in the denominators, which can simplify calculations and improve clarity in mathematical expressions.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domain. They are essential tools in solving and simplifying trigonometric equations.
In the solution to this exercise, several trigonometric identities are implicitly used:

• The reciprocal identities: \( \csc \theta = \frac{1}{\sin \theta} \) and \( \sec \theta = \frac{1}{\cos \theta} \)
• The quotient identity: \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)
• The Pythagorean identity for finding \( r = \sqrt{x^2 + y^2} \)

These identities allow us to find the remaining trigonometric values when only some are given, demonstrating their interconnectedness. Understanding and using trigonometric identities is key to mastering trigonometric functions.