Problem 17 - 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. $$(5,-12)$$

Step-by-Step Solution

Answer

\( \theta \) is in the fourth quadrant with trigonometric functions: \( \sin \theta = -\frac{12}{13} \), \( \cos \theta = \frac{5}{13} \), their reciprocals, and \( \tan \theta = -\frac{12}{5} \).

Step 1: Plot the Point

To start, plot the point (5, -12) on a Cartesian plane. Since 5 is positive and -12 is negative, this point is located in the fourth quadrant. The x-coordinate represents the horizontal distance, and the y-coordinate represents the vertical distance.

Step 2: Find the Reference Angle

The reference angle, \( \alpha \), is calculated in the standard position by forming a right triangle with the x-axis. Using the point (5, -12), find \( \alpha \) using the tangent function: \( \tan \alpha = \frac{|y|}{x} = \frac{12}{5} \). Thus, \( \alpha = \arctan \left( \frac{12}{5} \right) \). Note that this computes the angle by the absolute values only, indicating its position relative to the x-axis.

Step 3: Determine the Angle \( \theta \) in Standard Position

Angles in standard position are measured from the positive x-axis. Since the terminal side is in the fourth quadrant, \( \theta \) is calculated as \( 360^\circ - \alpha \).

Step 4: Calculate Hypotenuse

Using the Pythagorean theorem, calculate the hypotenuse \( r \) of the right triangle formed. \( r = \sqrt{x^2 + y^2} = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).

Step 5: Calculate Sine and Cosine

Using the coordinates and the hypotenuse: \( \sin \theta = \frac{y}{r} = \frac{-12}{13} \) and \( \cos \theta = \frac{x}{r} = \frac{5}{13} \).

Step 6: Calculate Other Trigonometric Functions

From \( \sin \theta \) and \( \cos \theta \), calculate: \( \tan \theta = \frac{y}{x} = \frac{-12}{5} \), \( \csc \theta = \frac{r}{y} = \frac{13}{-12} \), \( \sec \theta = \frac{r}{x} = \frac{13}{5} \), and \( \cot \theta = \frac{x}{y} = \frac{5}{-12} \). Remember to rationalize the denominators where needed.

Key Concepts

Standard Position

When discussing angles in trigonometry, the concept of "standard position" is essential. Standard position refers to placing an angle on a coordinate plane so that its vertex is at the origin \(0,0\), and its initial side lies along the positive x-axis.
This orientation makes it easier to measure and analyze the angle, as well as identify its quadrant where it lies based on the point given, such as \(5, -12\).

• The initial side always starts from the positive x-axis.
• The rotation direction determines whether the angle is positive (counterclockwise) or negative (clockwise).

For the angle \( \theta \) formed by the point \(5, -12\), the terminal side falls in the fourth quadrant because the x-coordinate is positive, and the y-coordinate is negative. This placement helps us understand the position of the angle better and aids in calculating functions like sine and cosine.

Reference Angle

The reference angle is a fundamental concept in trigonometry that helps simplify the computation of trigonometric functions. In essence, it is the smallest angle that the terminal side of an angle \(\theta\) makes with the x-axis.
This angle is always positive and less than \(90^\circ\) regardless of the actual angle's quadrant.
To find the reference angle \(\alpha\) for our angle with the terminal side passing through \(5, -12\):

• We create a right triangle with the x-axis as one side, using the tangent function: \( \tan \alpha = \frac{|y|}{x} = \frac{12}{5} \).
• The reference angle \(\alpha\) is therefore \(\arctan\left(\frac{12}{5}\right)\), representing the angle made with the positive x-axis.

This reference angle helps determine the specific trigonometric function values for angles in different quadrants, by considering just the acute angle, simplifying calculations.

Pythagorean Theorem

The Pythagorean theorem is pivotal in calculating distances in trigonometry. It is especially crucial when calculating the hypotenuse of a right triangle formed by the components of a given point on the coordinate plane.
This theorem states that in a right triangle, the square of the hypotenuse \(r\) is equal to the sum of the squares of the other two sides, \(x\) and \(y\).
For the point \(5, -12\), we apply the Pythagorean theorem to find the hypotenuse:

• The formula is: \[ r = \sqrt{x^2 + y^2} \]
• Plug in the values: \[ r = \sqrt{5^2 + (-12)^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \]

This calculation of \(r\) allows us to further calculate trigonometric functions such as sine and cosine, by providing the denominator for ratios based on the right triangle model.